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Mathematics of Poker
This fallacious belief that the coin's past history affects its future is fun to explore with further questions.
- Suppose I flip the coin three times and it comes up three heads, then I put it away. Tomorrow I take it out again. Is the probability still greater that a tail will occur? If the probability is greater, what happens if I wait a week or a month or a year?
- If after some elapsed time interval (day, week, month), the probabilities reset to 50-50, why does this happen?
- If there is no time interval that resets the probabilities to 50-50, how do I know it wasn't due for three heads in a row? After all, I may have just received the coin, so how do I know the previous owner hadn't flipped it five times and gotten all tails?
- How exactly has the probability changed of a tail occurring after three heads in a row? What will it be if just one head has appeared, or after five heads in row?
If my friend's belief is true, all these questions should have answers. The fact that all answers to these questions would be nonsense shows the belief is false.
Of course, a more profitable exercise would be to find a betting person who believes that a past sequence of coin flips affects the future. If this person is willing to pay better than even money on the outcome of a fourth head each time three heads appear in a row, you are assured of making money. Since past coin flips have no affect on future ones, the probabilities remain 50-50. Anyone willing to pay out better than even money on wagers against coin flips even selected coin flips that occur after streaks-will 10se over the long-run.
I relate the story of my friend's belief in coin flips because it is easy to see that the belief is wrong. However, many gamblers share the same wrong belief and don't realize it. They will behave as if the past does affect the future.
Examples of common behaviors:
- You have just lost seven hands in a row at blackjack. You decide to bet the table maximum on the eighth hand because it must be time for a win. After all, how often do you 10se eight hands in a row?
- You have received garbage poker hands for the last hour and decide that no matter what, you will bet heavily on the next hand because it must be time for a winning hand.
- You repeatedly play the same lottery numbers, believing that if those numbers haven't won yet, their time must be coming.
Each of these behaviors results from believing that "I'm due". Claiming that "I'm due" is just another way of claiming that the past affects the future.
There are more subtle expressions of this wrong thinking, such as believing that other players in the game affect the probabilities.
- Some poker players believe that the more players there are competing for a pot, the more likely it is that they will be dealt three-of-a-kind or better. It is easy to fall into this trap, since winning hands tend to be higher when there are more players at the table. Three of a kind is much more likely to be the winning hand when seven players compete, while with two players, a high pair is often enough to win. However, winning hands are higher, only because more hands are dealt, and to win, you have to beat more people. Whatever the number of players, the probability of you, an individual, receiving three-of-a-kind is always the same. Present events, such as the number of players who decide to fold early, do not after the fact affect the probabilities on the past event of the deal. Just as the past doesn't affect the future, the present does not affect the past.
- Some poker players believe in betting heavily during winning streaks and advise not to leave the table. This may be sound advice if the reasons for the winning streak have to do with being at a table filled with bad players. However, if the reason is a sudden streak of great cards, there is no reason to believe this has anything to do with the table, your seat, your opponents, the dealer, the day of the week, the color of your socks, or any factor, period. Good streaks happen just like bad streaks. There are no reasons; that is just the nature of a series of random events.
Make sure that your playing decisions are grounded in correct reasons. Analyze the reasons for your decisions. If the reason comes down to believing that the past affects the future, the reason is wrong.
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