Also it’s useful in arguments, you can offer a bet and force your opponent to either accept it or publicly update their beliefs.
The kind of update on your beliefs you might make may not necessarily be the kind of belief the bet is supposed to demonstrate, however. For instance, in your example, you believe that the coin is a fair coin. Someone flips it and says “If you think this is a fair coin, I’ll bet you that the coin came up heads”. Because he would probably offer you the bet if it came up heads, you should update on the belief that the coin came up heads this time. However, you shouldn’t update much, if at all, on the belief that the coin is fair.
But the bet is being presented as a test of your belief that the coin is fair. So the fact that you updated doesn’t actually indicate that you have changed your mind on the important aspect of the bet.
The kind of update on your beliefs you might make may not necessarily be the kind of belief the bet is supposed to demonstrate, however. For instance, in your example, you believe that the coin is a fair coin. Someone flips it and says “If you think this is a fair coin, I’ll bet you that the coin came up heads”. Because he would probably offer you the bet if it came up heads, you should update on the belief that the coin came up heads this time. However, you shouldn’t update much, if at all, on the belief that the coin is fair.
But the bet is being presented as a test of your belief that the coin is fair. So the fact that you updated doesn’t actually indicate that you have changed your mind on the important aspect of the bet.