Coin Game \ Magic Trick \ Maths: The Penney Ante Part 1 (Re: Derren Brown: How to Win the Lottery)
Hello, everybody! So–about last night. For people not in the UK, last Wednesday, a brilliant UK magician called Derren Brown predicted the national lottery numbers on live TV. And last night he explained how he did it. Now, I won’t go into that, because Derren gave a few explanations, and they were all nonsense. However, I did laugh when the following coin game came up as proof of psychic ability. Now, Derren did say that this was a maths effect, based on what he called “deep maths.” I’m going to explain that game. I’m going to explain that game, I’m going to tell you the optimal strategy for you to win, and I’m going to explain some of that “deep maths” behind it. And hopefully you’ll be able to try this out on your friends at home. Now, the game is called “Penney Ante.” It was devised in 1969 by a mathematician called Walter Penney. It works like this– player one goes first. He has to predict three coin tosses. So these are the possibilities. It could be heads-heads-heads, heads-heads-tails, heads-tails-heads, heads-tails-tails, tails-heads-heads, tails-heads-tails, tails-tails-heads, or tails-tails-tails. So there are eight possibilities. Player one picks one of these as his prediction. Player two then makes his prediction, and the prediction that appears first in a run of coin tosses wins. Now, since these are equally likely, it sounds like a fair game. But, if you go second, there’s a strategy to help you increase your chances of winning. For example, if player one picks heads-heads-heads, player two should pick tails-heads-heads. Because if this is a run of coin tosses– let’s say this is the first time heads-heads-heads appears in this run. Then the preceding coin must be tails. If it was heads, then this wouldn’t be the first time heads-heads-heads appears in that run. So the preceding coin must be tails, which means player two’s choice, tails-heads-heads, must appear first. And player two wins! Now, the only exception to this is if the first three coin tosses are heads-heads-heads. Now, the chances of that happening are one in eight, so the probability that player two is going to win is seven in eight, which are pretty good odds! Now, given player one’s choice, this is a table that tells you what player two should do. For example, if player one picks heads-tails-tails, then player two should pick heads-heads-tails. And if he does, he’s going to win with a probability of two-thirds. That’s about 67%. Now, there’s an easy way to remember this table. What you do is you take player one’s choice, you make a copy of his middle coin, and you flip it over, and you put it at the front. So if player one picks tails-heads-tails, you make a copy of the middle coin, heads, flip it over so it’s tails, put it at the front. And so player two’s choice is tails-tails-heads. And if he does that, the probability that he will win is three-quarters– 75%. Pretty good! Now, this is where most explanations stop. After the break, I’m going to go a bit further. I’m going to show you how we work out the best strategy, and those chances of winning.