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How that Coin Toss thing Works! Okay people, this is pretty heavy in probability, so if you don't understand any of it, you won't be the only one. Have a go at following what I'm saying though, but if you don't get it, don't worry. Okay, here we go. Take the simplest example from the other page.
Their bet: HHH Let's start with the probability of their selection winning. What they need for their selection to win, is a head, followed by a head, followed by a head; which has a probability of (0.5 x 0.5 x 0.5), or one eighth. Now let's look at the probability of your bet winning. In theorey, the probabilities are equal, and yours should be one eighth as well, but actually, it's not. Let's look at it in order. Say, for one of the first three throws, you throw a tail (any throw.) The probability of throwing a tail in the first throw is 0.5. Supposing this one comes out a head, the probability of throwing a tail on the second throw is 0.5, and the same for the third throw. As soon as you throw a tail, they're completely out of the competition, because for them to win, they need 3 heads in a row, but as soon as you throw a tail, if you throw 2 heads next, which are the first 2 of their selection, you've won, and they still have 1 head to go. Even if you throw H,H,T,H, if you throw another head, you've won, and there is absolutely no way they can win, now that you've thrown a tail. From this, we can see that the probability of their winning is one eigth, but the probability of you winning is one half, which happens to be 4:1 on your side. (ie: that the probability of you winning is 4/5.) Please make sure you get this much before you go onto the next example. Now let's take a slightly more complicated example.
Their bet: THH Suppose you continually toss the coin, and keep going until one or the other of you wins. Let's say that we are looking at any three tosses, not the first three. The probability of their winning, as was demonstrated in the last paragraph, is one eighth. Now, for you to win, all you need is for the previous toss to be a tail, because as soon as the previous toss is a tail, the first two letters of their bet become the last two of yours, and you've won. The probability of the previous toss being a tail is 0.5, so the probability of you winning this bet is again 4/5. Let's look at the last two possible examples.
Their bet: TTH (HTH) These sets of bets have exactly the same explanation as the last set. The probability of their winning is one eighth. For you to win, all you need is for the previous toss to be a head (head), which has the probability of 0.5. This makes the probability of you winning 4/5 in both cases.
This page covers all the possible outcomes of a toss-up of this kind, because, of course, if you reverse the heads and tails you get all possible combinations. I only just understand it myself, but I've tried to explain it as best I can here, so email me if you have any problems.
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