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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the generalised mathematical concept, see <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale (probability theory)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Martingale_(betting_system)" title="Special:EditPage/Martingale (betting system)">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Martingale%22+betting+system">"Martingale" betting system</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Martingale%22+betting+system+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Martingale%22+betting+system&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Martingale%22+betting+system+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Martingale%22+betting+system">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Martingale%22+betting+system&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A <b>martingale</b> is a class of <a href="/wiki/Betting_strategy" title="Betting strategy">betting strategies</a> that originated from and were popular in 18th-century <a href="/wiki/France" title="France">France</a>. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy is an instantiation of the <a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg paradox</a>. </p><p>Since a gambler will <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> eventually <a href="/wiki/Coin_flipping" title="Coin flipping">flip heads</a>, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler has infinite wealth, and the <a href="/wiki/Exponential_growth" title="Exponential growth">exponential growth</a> of the bets can bankrupt unlucky gamblers who choose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the <a href="/wiki/Gambling_mathematics#House_advantage_or_edge" title="Gambling mathematics">house's edge</a>. Additionally, as the likelihood of a string of consecutive losses is higher than common intuition suggests, martingale strategies can bankrupt a gambler quickly.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The martingale strategy has also been applied to <a href="/wiki/Roulette" title="Roulette">roulette</a>, as the probability of hitting either red or black is close to 50%.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Intuitive_analysis">Intuitive analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=1" title="Edit section: Intuitive analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win–loss outcomes of each bet are <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed random variables</a>, an assumption which is valid in many realistic situations.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative. </p><p>The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).<sup id="cite_ref-mitzenmacherupfal_4-0" class="reference"><a href="#cite_note-mitzenmacherupfal-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It is only with unbounded wealth, bets <i>and</i> time that it could be argued that the martingale becomes a <a href="/wiki/Winning_strategy" class="mw-redirect" title="Winning strategy">winning strategy</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_analysis">Mathematical analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=2" title="Edit section: Mathematical analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the <a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">optional stopping theorem</a>.<sup id="cite_ref-mitzenmacherupfal_4-1" class="reference"><a href="#cite_note-mitzenmacherupfal-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_analysis_of_a_single_round">Mathematical analysis of a single round</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=3" title="Edit section: Mathematical analysis of a single round"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler "resets" and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value of one round. </p><p>Let <i>q</i> be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). Let <i>B</i> be the amount of the initial bet. Let <i>n</i> be the finite number of bets the gambler can afford to lose. </p><p>The probability that the gambler will lose all <i>n</i> bets is <i>q</i><sup><i>n</i></sup>. When all bets lose, the total loss is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}B\cdot 2^{i-1}=B(2^{n}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>B</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>B</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}B\cdot 2^{i-1}=B(2^{n}-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22403a7f70e3932e172d93c84b527bb2afe22632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.303ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}B\cdot 2^{i-1}=B(2^{n}-1)}"></span></dd></dl> <p>The probability the gambler does not lose all <i>n</i> bets is 1 − <i>q</i><sup><i>n</i></sup>. In all other cases, the gambler wins the initial bet (<i>B</i>.) Thus, the <a href="/wiki/Expected_value" title="Expected value">expected</a> profit per round is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-q^{n})\cdot B-q^{n}\cdot B(2^{n}-1)=B(1-(2q)^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>B</mi> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mi>B</mi> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-q^{n})\cdot B-q^{n}\cdot B(2^{n}-1)=B(1-(2q)^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c78ddc6565678b3fe2df6f9cbac2b04000a722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.262ex; height:2.843ex;" alt="{\displaystyle (1-q^{n})\cdot B-q^{n}\cdot B(2^{n}-1)=B(1-(2q)^{n})}"></span></dd></dl> <p>Whenever <i>q</i> > 1/2, the expression 1 − (2<i>q</i>)<sup><i>n</i></sup> < 0 for all <i>n</i> > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average loss. </p><p>Suppose a gambler has a 63-unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking <i>k</i> as the number of preceding consecutive losses, the player will always bet 2<sup>k</sup> units. </p><p>With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet. </p><p>With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued. </p><p>In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19)<sup>6</sup> = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19)<sup>6</sup> = 97.8744%. </p><p>The expected amount won is (1 × 0.978744) = 0.978744.<br /> The expected amount lost is (63 × 0.021256)= 1.339118.<br /> Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 . </p><p>In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64. Assuming <i>q</i> > 1/2 (it is a real casino) and he may only place bets at even odds, his best strategy is <b>bold play</b>: at each spin, he should bet the smallest amount such that if he wins he reaches his target immediately, and if he does not have enough for this, he should simply bet everything. Eventually he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible in this circumstance.<sup id="cite_ref-dubinssavage_5-0" class="reference"><a href="#cite_note-dubinssavage-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds (but still with the same expected loss of 10/19 of the stake at each bet), and can only place one bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely bold play.<sup id="cite_ref-shepp_6-0" class="reference"><a href="#cite_note-shepp-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_mathematical_analysis">Alternative mathematical analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=4" title="Edit section: Alternative mathematical analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The previous analysis calculates <i>expected value</i>, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll? </p><p>As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll. </p><p>In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown that since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. In fact, while the chance of losing 6 times in a row in 6 plays is a relatively low 1.8% on a single-zero wheel, the probability of losing 6 times in a row (i.e. encountering a streak of 6 losses) <i>at some point</i> during a string of 200 plays is approximately 84%. Even if the gambler can tolerate betting ~1,000 times their original bet, a streak of 10 losses in a row has an ~11% chance of occurring in a string of 200 plays. Such a loss streak would likely wipe out the bettor, as 10 consecutive losses using the martingale strategy means a loss of 1,023x the original bet. </p><p>These unintuitively risky probabilities raise the bankroll requirement for "safe" long-term martingale betting to infeasibly high numbers. To have an under 10% chance of failing to survive a long loss streak during 5,000 plays, the bettor must have enough to double their bets for 15 losses. This means the bettor must have over 65,500 (2^15-1 for their 15 losses and 2^15 for their 16th streak-ending winning bet) times their original bet size. Thus, a player making 10 unit bets would want to have over 655,000 units in their bankroll (and still have a ~5.5% chance of losing it all during 5,000 plays). </p><p>When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This intuitive belief is sometimes referred to as the <a href="/wiki/Representativeness_heuristic" title="Representativeness heuristic">representativeness heuristic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Anti-martingale">Anti-martingale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=5" title="Edit section: Anti-martingale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach, also known as the reverse martingale, instead increases bets after wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or a "hot hand", while reducing losses while "cold" or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning "streaks" is merely an example of <a href="/wiki/Gambler%27s_fallacy" title="Gambler's fallacy">gambler's fallacy</a>, and the anti-martingale strategy fails to make any money.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid 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pattern in which the participant takes on increasingly greater risk</li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg paradox</a> – Paradox involving a game with repeated coin flipping</li> <li><a href="/wiki/Sunk_cost_fallacy" class="mw-redirect" title="Sunk cost fallacy">Sunk cost fallacy</a> – Cost that has already been incurred and cannot be recovered<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Martingale_(betting_system)&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://spinsavvy.co.uk/understanding-the-martingale-strategy-in-gambling/">"Mastering the Martingale Strategy in Gambling"</a>. 2024-09-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-11-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mastering+the+Martingale+Strategy+in+Gambling&rft.date=2024-09-30&rft_id=https%3A%2F%2Fspinsavvy.co.uk%2Funderstanding-the-martingale-strategy-in-gambling%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams2021" class="citation book cs1">Williams, Leighton Vaughan (2021-09-15). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YHU_EAAAQBAJ&dq=The+martingale+strategy+has+also+been+applied+to+roulette,+as+the+probability+of+hitting+either+red+or+black+is+close+to+50%25.&pg=PA152"><i>Probability, Choice, and Reason</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-000-45887-9" title="Special:BookSources/978-1-000-45887-9"><bdi>978-1-000-45887-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%2C+Choice%2C+and+Reason&rft.pub=CRC+Press&rft.date=2021-09-15&rft.isbn=978-1-000-45887-9&rft.aulast=Williams&rft.aufirst=Leighton+Vaughan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYHU_EAAAQBAJ%26dq%3DThe%2Bmartingale%2Bstrategy%2Bhas%2Balso%2Bbeen%2Bapplied%2Bto%2Broulette%2C%2Bas%2Bthe%2Bprobability%2Bof%2Bhitting%2Beither%2Bred%2Bor%2Bblack%2Bis%2Bclose%2Bto%2B50%2525.%26pg%3DPA152&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOttavianiSørensen2010" class="citation journal cs1">Ottaviani, Marco; Sørensen, Peter Norman (2010). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25760376">"Noise, Information, and the Favorite-Longshot Bias in Parimutuel Predictions"</a>. <i>American Economic Journal: Microeconomics</i>. <b>2</b> (1): 58–85. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1945-7669">1945-7669</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25760376">25760376</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Economic+Journal%3A+Microeconomics&rft.atitle=Noise%2C+Information%2C+and+the+Favorite-Longshot+Bias+in+Parimutuel+Predictions&rft.volume=2&rft.issue=1&rft.pages=58-85&rft.date=2010&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25760376%23id-name%3DJSTOR&rft.issn=1945-7669&rft.aulast=Ottaviani&rft.aufirst=Marco&rft.au=S%C3%B8rensen%2C+Peter+Norman&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25760376&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> <li id="cite_note-mitzenmacherupfal-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-mitzenmacherupfal_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mitzenmacherupfal_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichael_MitzenmacherEli_Upfal2005" class="citation cs2">Michael Mitzenmacher; Eli Upfal (2005), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151013024650/https://books.google.com/books?id=0bAYl6d7hvkC&pg=PA298"><i>Probability and computing: randomized algorithms and probabilistic analysis</i></a>, Cambridge University Press, p. 298, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83540-4" title="Special:BookSources/978-0-521-83540-4"><bdi>978-0-521-83540-4</bdi></a>, archived from <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0bAYl6d7hvkC&pg=PA298">the original</a> on October 13, 2015</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+computing%3A+randomized+algorithms+and+probabilistic+analysis&rft.pages=298&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-83540-4&rft.au=Michael+Mitzenmacher&rft.au=Eli+Upfal&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0bAYl6d7hvkC%26pg%3DPA298&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span> </span> </li> <li id="cite_note-dubinssavage-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-dubinssavage_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLester_E._DubinsLeonard_J._Savage1965" class="citation cs2"><a href="/wiki/Lester_Dubins" title="Lester Dubins">Lester E. 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Savage</a> (1965), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kt9QAAAAMAAJ"><i>How to gamble if you must: inequalities for stochastic processes</i></a>, McGraw Hill</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+to+gamble+if+you+must%3A+inequalities+for+stochastic+processes&rft.pub=McGraw+Hill&rft.date=1965&rft.au=Lester+E.+Dubins&rft.au=Leonard+J.+Savage&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dkt9QAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> <li id="cite_note-shepp-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-shepp_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLarry_Shepp2006" class="citation cs2">Larry Shepp (2006), <a rel="nofollow" class="external text" href="https://www.worldscientific.com/doi/abs/10.1142/9789812772558_0010"><i>Bold play and the optimal policy for Vardi's casino, pp 150–156 in: Random Walk, Sequential Analysis and Related Topics</i></a>, World Scientific, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789812772558_0010">10.1142/9789812772558_0010</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bold+play+and+the+optimal+policy+for+Vardi%27s+casino%2C+pp+150%E2%80%93156+in%3A+Random+Walk%2C+Sequential+Analysis+and+Related+Topics&rft.pub=World+Scientific&rft.date=2006&rft_id=info%3Adoi%2F10.1142%2F9789812772558_0010&rft.au=Larry+Shepp&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fdoi%2Fabs%2F10.1142%2F9789812772558_0010&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin2009" class="citation web cs1">Martin, Frank A. (February 2009). <a rel="nofollow" class="external text" href="http://wizardofodds.com/image/ask-the-wizard/streaks.pdf">"What were the Odds of Having Such a Terrible Streak at the Casino?"</a> <span class="cs1-format">(PDF)</span>. WizardOfOdds.com<span class="reference-accessdate">. Retrieved <span class="nowrap">31 March</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+were+the+Odds+of+Having+Such+a+Terrible+Streak+at+the+Casino%3F&rft.pub=WizardOfOdds.com&rft.date=2009-02&rft.aulast=Martin&rft.aufirst=Frank+A.&rft_id=http%3A%2F%2Fwizardofodds.com%2Fimage%2Fask-the-wizard%2Fstreaks.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://capital.com/the-martingale-approach-and-averaging-down">"Martingale Strategy and Averaging Down | What You Need to Know"</a>. <i>capital.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=capital.com&rft.atitle=Martingale+Strategy+and+Averaging+Down+%7C+What+You+Need+to+Know&rft_id=https%3A%2F%2Fcapital.com%2Fthe-martingale-approach-and-averaging-down&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMartingale+%28betting+system%29" class="Z3988"></span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Africa</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gambling_in_Angola" title="Gambling in Angola">Angola</a></li> <li><a href="/wiki/Gambling_in_Nigeria" title="Gambling in Nigeria">Nigeria</a></li> <li><a href="/wiki/Gambling_in_South_Africa" title="Gambling in South Africa">South Africa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Asia</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gambling_in_Bangladesh" title="Gambling in Bangladesh">Bangladesh</a></li> <li><a href="/wiki/Gambling_in_Cambodia" title="Gambling in Cambodia">Cambodia</a></li> <li><a href="/wiki/Gambling_in_China" title="Gambling in China">China</a> <ul><li><a href="/wiki/Gambling_in_Hong_Kong" title="Gambling in Hong Kong">Hong Kong</a></li> <li><a href="/wiki/Gambling_in_Macau" title="Gambling in Macau">Macau</a></li></ul></li> <li><a href="/wiki/Gambling_in_India" title="Gambling in India">India</a></li> <li><a href="/wiki/Gambling_in_Japan" title="Gambling in Japan">Japan</a></li> <li><a href="/wiki/Gambling_in_Pakistan" title="Gambling in Pakistan">Pakistan</a></li> <li><a href="/wiki/Gambling_in_the_Philippines" title="Gambling in the Philippines">Philippines</a> <ul><li><a href="/wiki/Gambling_in_Metro_Manila" title="Gambling in Metro Manila">Manila</a></li></ul></li> <li><a href="/wiki/Gambling_in_Russia" title="Gambling in Russia">Russia</a></li> <li><a href="/wiki/Gambling_in_Singapore" title="Gambling in Singapore">Singapore</a></li> <li><a href="/wiki/Gambling_in_Taiwan" title="Gambling in Taiwan">Taiwan</a></li> <li><a href="/wiki/Gambling_in_Thailand" title="Gambling in Thailand">Thailand</a></li> <li><a href="/wiki/Gambling_in_Turkey" title="Gambling in Turkey">Turkey</a></li> <li><a href="/wiki/Gambling_in_Vietnam" title="Gambling in Vietnam">Vietnam</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Europe</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gambling_in_Estonia" title="Gambling in Estonia">Estonia</a></li> <li><a href="/wiki/Gambling_in_France" title="Gambling in France">France</a></li> <li><a href="/wiki/Gambling_in_Italy" title="Gambling in Italy">Italy</a></li> <li><a href="/wiki/Gambling_in_Norway" title="Gambling in Norway">Norway</a></li> <li><a href="/wiki/Gambling_in_Russia" title="Gambling in Russia">Russia</a></li> <li><a href="/wiki/Gambling_in_Turkey" title="Gambling in Turkey">Turkey</a></li> <li><a href="/wiki/Gambling_in_Ukraine" title="Gambling in Ukraine">Ukraine</a></li> <li><a href="/wiki/Gambling_in_the_United_Kingdom" title="Gambling in the United Kingdom">United Kingdom</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">North America</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gambling_in_Mexico" title="Gambling in Mexico">Mexico</a></li> <li><a href="/wiki/Gambling_in_the_United_States" title="Gambling in the United States">United States</a></li> <li>Canada <ul><li><a href="/wiki/Gambling_in_Quebec" title="Gambling in Quebec">Quebec</a></li> <li><a href="/wiki/Ontario_Lottery_and_Gaming_Corporation" title="Ontario Lottery and Gaming Corporation">Ontario</a></li> <li><a href="/wiki/British_Columbia_Lottery_Corporation" title="British Columbia Lottery Corporation">BC</a></li> <li><a href="/wiki/Western_Canada_Lottery_Corporation" title="Western Canada Lottery Corporation">Western Canada</a></li> <li><a href="/wiki/Atlantic_Lottery_Corporation" title="Atlantic Lottery Corporation">Atlantic Canada</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Oceania</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gambling_in_Australia" title="Gambling in Australia">Australia</a></li> <li><a href="/wiki/Gambling_in_New_Zealand" title="Gambling in New Zealand">New Zealand</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">South America</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Gambling_in_Brazil" title="Gambling in Brazil">Brazil</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Gambling_games" title="Category:Gambling games">Games</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Casino_game" title="Casino game">Casino game</a></li> <li><a href="/wiki/Game_of_chance" title="Game of chance">Game of chance</a></li> <li><a href="/wiki/Game_of_skill" title="Game of skill">Game of skill</a></li> <li><a href="/wiki/List_of_bets" title="List of bets">List of bets</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Issues</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gaming_law" title="Gaming law">Gaming law</a></li> <li><a href="/wiki/Problem_gambling" title="Problem gambling">Problem gambling</a></li> <li><a href="/wiki/Video_game_addiction" title="Video game addiction">Video game addiction</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" 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