# Odds of 5 tails in a row in 20 throws? **URL:** \ **Category:** Factual Questions\ **Created:** [October 30, 2004, 7:55pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057 "2004-10-30T19:55:05Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![PosterChild](https://avatars.discourse-cdn.com/v4/letter/p/e274bd/32.png) [@PosterChild](https://boards.straightdope.com/u/PosterChild)\ **Post date:** [October 30, 2004, 7:55pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/1 "2004-10-30T19:55:05Z")
A friend of mine was trying to calculate the odds of loosing it all playing Martingale’s strategy at roulette. There have been a number of threads about why is’s a losing strategy, but I was trying to figure out how to calculate her chances of loosing 5, 6 or 7 times (the number of loses she could cover based on her first bet) in a row if she played 20 times or so. By analogy, I tried to use flipping a coin (though that’s better odds than the roulette wheel), but factorials and stuff are not my strong point. So, if you flipped a coin 20 times, what are your chances of getting a string of 5, 6 or 7 tails? Or for that matter, what are the odds of 5, 6 or 7 losses if you’re playing red or black on a roulette table? PC ---
**Author:** ![PosterChild](https://avatars.discourse-cdn.com/v4/letter/p/e274bd/32.png) [@PosterChild](https://boards.straightdope.com/u/PosterChild)\ **Post date:** [October 30, 2004, 8:09pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/2 "2004-10-30T20:09:40Z")
> [@PosterChild](#): > > A friend of mine was trying to calculate the odds of loosing it all playing Martingale’s strategy at roulette. There have been a number of threads about why is’s a losing strategy, but I was trying to figure out how to calculate his chances of loosing 5, 6 or 7 times (the number of loses he could cover based on her first bet) in a row if he played 20 times or so. By analogy, I tried to use flipping a coin (though that’s better odds than the roulette wheel), but factorials and stuff are not my strong point. > > So, if you flipped a coin 20 times, what are your chances of getting a string of 5, 6 or 7 tails? Or for that matter, what are the odds of 5, 6 or 7 losses if you’re playing red or black on a roulette table? > > PC I just realized that since he’d walk away after his limit was reached, it isn’t necessarily 20 throws, but up to 20 depending on the chance of first getting to the string of tails/losses. ---
**Author:** ![aahala](https://avatars.discourse-cdn.com/v4/letter/a/a88e4f/32.png) [@aahala](https://boards.straightdope.com/u/aahala)\ **Post date:** [October 30, 2004, 8:58pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/3 "2004-10-30T20:58:55Z")
> [@PosterChild](#): > > So, if you flipped a coin 20 times, what are your chances of getting a string of 5, 6 or 7 tails? Or for that matter, what are the odds of 5, 6 or 7 losses if you’re playing red or black on a roulette table? > > PC The expected number of “r-runs” of one side of a coin in n tosses is: n/2\*\*(r+2) Your odds on a particular spin at roulette is slightly less than 50/50 - the wheel usually has one or two slots, “0” and “00” only are house wins, with 18 red and 18 black ones as well. Longer strings of losses are a lot more likely than in coin flipping because of this. ---
**Author:** ![PosterChild](https://avatars.discourse-cdn.com/v4/letter/p/e274bd/32.png) [@PosterChild](https://boards.straightdope.com/u/PosterChild)\ **Post date:** [October 30, 2004, 9:09pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/4 "2004-10-30T21:09:51Z")
Does it change if you stop after an “r-run” and don’t continue the tosses? ---
**Author:** ![aahala](https://avatars.discourse-cdn.com/v4/letter/a/a88e4f/32.png) [@aahala](https://boards.straightdope.com/u/aahala)\ **Post date:** [October 30, 2004, 9:20pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/5 "2004-10-30T21:20:40Z")
> [@PosterChild](#): > > Does it change if you stop after an “r-run” and don’t continue the tosses? Does WHAT change after you stop playing? Your attitude? I guess you could say the chances of further winning(or losing) go to zero.😃 ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [October 30, 2004, 9:26pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/6 "2004-10-30T21:26:16Z")
The probability of 5 in a row is 1/2[sup]5[/sup] or 1 in 32. I don’t think that the hypothetical total number of tries you intend to make enters into it but I’m willing to be convinced otherwise. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [October 30, 2004, 9:29pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/7 "2004-10-30T21:29:05Z")
On thinking it over, if you played a long series of of tries and broke it up into sets of 20 tries each wouldn’t the probability that any particular 20 try set contained 5 in a row be 20/32? ---
**Author:** ![Roadfood](https://avatars.discourse-cdn.com/v4/letter/r/9fc348/32.png) [@Roadfood](https://boards.straightdope.com/u/Roadfood)\ **Post date:** [October 30, 2004, 9:53pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/8 "2004-10-30T21:53:43Z")
> [@David Simmons](#): > > The probability of 5 in a row is 1/2[sup]5[/sup] or 1 in 32. > > I don’t think that the hypothetical total number of tries you intend to make enters into it but I’m willing to be convinced otherwise. A moment’s thought should convince you: Scenerio 1: you flip a coin five times. Scenerio 2: you flip a coin a hundred times. Isn’t it obvious that the odds of getting five heads in a row somewhere in a sequence of a hundred flips is better than the odds of getting five heads in a row in a seqence of five flips? In five flips, there is only one seqence of five, one chance to see five heads; in a hundred flips, there are 96 sequences of five, 96 chances to see five heads. Scenerio 3: you flip a coin a million times. The odds of getting five heads in a row somewhere during those million flips (999,996 sequences of five) has gotta be greater than the odds of getting five heads in a row sometime during a hundred flips (96 sequences of five). The fact that you intend to stop as soon as you hit those five heads in a row is irrelevant to the calculation of what the odds are of five heads in a row sometime in n flips. ---
**Author:** ![EllisDee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ellisdee/32/4531_2.png) [@EllisDee](https://boards.straightdope.com/u/EllisDee)\ **Post date:** [October 30, 2004, 10:17pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/9 "2004-10-30T22:17:07Z")
> [@aahala](#): > > Your odds on a particular spin at roulette is slightly less than 50/50 - the wheel usually has one or two slots, “0” and “00” only are house wins, with 18 red and 18 black ones as well. Longer strings of losses are a lot more likely than in coin flipping because of this. Slightly? Try enormously. The house edge is 5.26%, which makes it a sucker bet. The OP’s friend would be well served trying her martingale on the Don’t Pass line in craps for several reasons. First, the house edge is only 1.40%, which is close to the most you would normally want to tolerate. Second, it takes longer in real time, on average, to resolve a point in craps than to spin a roulette wheel once, which means the inevitable loss of all money will take longer. And third, you have complete dominion over your space in craps, whereas in roulette it’s akin to feeding time at the zoo when it’s time to place your bets. To address the OP, first off the “20 spins” concept is flawed. You don’t care about spins, you care about series. Whether it takes 1 spin, 4, or 20 to finally get a winner, all those spins equate to a single wager, and should therefore be considered as a single entity. “Series” works as well as any other word for this. So, what’s the chance of a single series failing? Let’s see, using the Martingale, and capping (as the OP did) at 5, 6, or 7, you can figure out the single series chance pretty easily. Chance to lose roulette is 20/38. Chance to lose 5 times in a row? (20/38)[sup]5[/sup]. nth times? (20/38)[sup]n[/sup] Let’s go with 6, because it’s in the middle, and also this way she can lose twice the money she would have had she capped losses at 5. 😉 (20/38)[sup]6[/sup] = 2.13% So let’s say we’re talking about a $5 wager, doubled after every loss, up to six losses max. Lose 5, 10, 20, 40, 80, 160. This is common, as bringing $320 to the casino is about right. You’ve got a 2.13% chance to lose $315, and 97.87% chance to win $5. How many did you say? 20 spins? I happen to know that 50/50 martingale train(wreck)s are resolved, on average, approximately every 3.5 trials or so. So that 20 spins would involve about 6 series, which would net you $30, but also involve a chance to win (to the 6th power when discussing 6 wins in a row) of 87.91% So there’s your chances, assuming you meant 20 spins, and not series. 87.91% chance to win $30, 12.09% chance to lose $315. If you meant 20 series, so that you could win $60, which IMO is much more likely, you’re talking about a 65.07% chance to win 20 martingale series’ in a row. In short, the martingale train will have over a 1 in 3 chance to wreck in 20 resolutions. Allow me to attempt to pinpoint exactly what the fallacy of the martingale is. I’ll just quote myself from [this thread](http://boards.straightdope.com/sdmb/showthread.php?t=263949): > [@Me](#): > > Let’s assume 50-50 wagers. Say, a coin flip. > > What’s the chance to lose 3 in a row? 12.5% > > So I lose 2 in a row. Now I should bump, because there’s a 100 - 12.5 = 87.5% chance to win, right? > > Wrong. You just lost 2 in a row, which in itself only has a 25% chance to occur. That means that you already realized the low percentage chance you are trying to capitalize on. As in, you missed your chance. > > Common sense tells you that losing 3 in a row only has a 12.5% of happening, so you should be 87.5% likely to win. But you’re not. Remember that the first 2 losses only has a 25% chance to happen in the first place. 12.5% is, not surprisingly, exactly half of 25%, keeping your current odds squarely at 50-50. > > Therefore, once you lose 2 in a row, you have painted yourself into a 25% total probability zone: 12.5% of the time you win, 12.5% of the time you lose, and 75% of the time you never got here in the first place. I sincerely hope that helps. Nobody should suffer the anguish of having to place their final bet in a martingale train. ---
**Author:** ![EllisDee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ellisdee/32/4531_2.png) [@EllisDee](https://boards.straightdope.com/u/EllisDee)\ **Post date:** [October 30, 2004, 10:21pm UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/10 "2004-10-30T22:21:20Z")
> [@Ellis Dee](#): > > If you meant 20 series, so that you could win **$60** Too many 6s in my post. That should be “20 series, so that you could win **$100** ” ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 31, 2004, 12:13am UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/11 "2004-10-31T00:13:39Z")
> [@aahala](#): > > The expected number of “r-runs” of one side of a coin in n tosses is: > > n/2\*\*(r+2) > > Your odds on a particular spin at roulette is slightly less than 50/50 - the wheel usually has one or two slots, “0” and “00” only are house wins, with 18 red and 18 black ones as well. Longer strings of losses are a lot more likely than in coin flipping because of this. Is that ‘\*\*’ as in exponentiation? ---
**Author:** ![aahala](https://avatars.discourse-cdn.com/v4/letter/a/a88e4f/32.png) [@aahala](https://boards.straightdope.com/u/aahala)\ **Post date:** [October 31, 2004, 4:35am UTC](https://boards.straightdope.com/t/odds-of-5-tails-in-a-row-in-20-throws/272057/12 "2004-10-31T04:35:16Z")
> [@ultrafilter](#): > > Is that ‘\*\*’ as in exponentiation? Yes.