debunking martingale

Odds of losing (or winning) a 50-50 x times in a row is simply 0.5^x

So 5 is 0.5^5 = 1/32
10 is 0.5^10 = 1/1024 etc...

Games of Chance
Have you ever seen have a streak of bad luck in a game of chance? They often become aggressive players, believing "I'm due!" These people make a common mistake, believing that the past affects odds in the future. It usually doesn't.

The Odds of a Coin Toss

Let's start with the simplest game of chance, a fair coin toss. On any given coin toss, there's a 1 in 2 chance that heads will come up, and there's also a 1 in 2 chance that tails will come up.

What are the odds that heads will come up twice in a row? It's 1 in (2 * 2). 1 in 4. The odds that heads will come up 3 times in a row? 1 in 8. 4 times? 1 in 16. 5 times? 1 in 32. You get the idea.

So imagine a gambler has been betting tails each time. Remember, tails never fails!

First toss, heads comes up. The gambler loses.

Second toss, heads comes up. The gambler loses again.

Third toss, heads comes up. The gambler loses again.

Fourth toss, heads comes up. The gambler loses again.

Fifth toss, heads comes up. The gambler loses again.

Chance Analysis

First, the gambler has had terrible "luck". As we know, the odds of the gambler losing 5 straight coin tosses is 1 in 32. (Note: The gambler's odds are the same whether he chose heads, tails, or any combination of the two.) The gambler knows the odds of losing 5 times in a row is 1 in 32, and he isn't pleased.

But the gambler also knows, that there's only a 1 in 64 chance of losing 6 coin tosses in a row. He's due! He decides to bet even more thinking that the 1 in 64 chance will never happen.

Is the Gambler Correct?

The gambler is making a common mistake. The coin came up heads 5 times in a row? It will never come up a 6th time. There's only a 1 in 64 chance of heads coming up on 6 times in a row.

Is the gambler correct? What are the odds that the 6th coin toss will be heads?

True Odds

The gambler is wrong. The previous 5 coin tosses have no bearing on the 6th toss. Each is an independent trial. Like the first toss, the odds of the 6th toss being heads is 1 in 2. The gambler thinks he's due, but he's no more due than he was on the first toss.

Conclusion

Now you know the truth about the odds behind a hot or cold streak. Next time you hear somebody say, "I'm due," you'll know that they really aren't! Don't get caught making the same mistake! by Scott Schlimmer