Gambler’s Fallacy

Because I have a long history of playing Texas Hold ‘Em No Limit Poker, I have always been intrigued by strategies and concepts surrounding the game and gambling in general.  One interesting concept is the Gambler’s Fallacy. 

The Gambler’s Fallacy refers to a person’s failure to recognize the real probability of an event.  It is most commonly explained as a person believing that deviations from expected results in successive independent events of a random process will result in future deviations in the opposite direction being more likely.  A common example of this is coin tossing.  If a coin is tossed 10 times in a row and lands on heads every time, then what are the chances that the 11th toss will result in tails?  Because each coin toss is an independent event, the probability of the next toss landing on tails is 1/2.  The fallacy of the gambler is that he believes because there have been 10 heads in a row, that a tails is more likely now. 

While each coin toss is an independent event, the gambler may actually be right in the long run.  While he may not be able to predict any one coin toss, according to the law of large numbers the results from a large number of trials should be closer to the expected result and will tend to become closer and closer to the expected result as more trials are performed.  The expected result for heads is 50% and tails is 50%.  This means that if the experimental result is heads 80% of the time after 100 trials, then there should eventually be a run of tails that balances it out and makes the experimental results closer to the expected results of 50%.  As you approach infinite trials, the experimental results should be very, very close to 50%.

How does this apply to casinos and gambling? 

When you gamble at a casino you may win or lose regardless of the chances of you winning.  If there is a 25% chance that you will win a particular gamble, and you win twice in a row then that does not affect the casino.  An individual experiences gambling at a casino as a very small sample size.  They will have deviations from the expected result simply because of the fact that they are using a small sample size.  The casino however can afford to have some individuals go on winning streaks because they know that other individuals will have losing streaks.  The casino experiences gambling using a very large sample size, so according to the law of large numbers the casino can expect the expected results of a game as they run more and more trials.  That means that while you may win 75% of the time in a game with an expected result of 25% wins, the casino does not worry because they know that in the long run the overall trials from all customers will be close to 25%. 

Although I can’t prove it, just logically, it seems to me that Gambler’s fallacy only applies to small sample sizes.  The gambler usually has a small sample size to work with as compared to the casino.  I suppose this is why it’s not called the Casino’s Fallacy. 

How this applies to Poker

If you have an 80% chance of winning a hand in poker, then you should always play it because 4 out of 5 times you will win.  This does not mean that you will win 4 times if you play the hand 5 times.  In fact, you may lose five hands in a row with an 80% chance of winning because of the small sample size.  However, if you play a large number of these hands then you can expect to eventually win about 80% of these hands due to the law of large numbers.  

Large sample sizes play a big role in how players make decision in No Limit Texas Hold ‘Em Poker.  Pot odds and implied pot odds only work in the long run because you can lose any one hand.  Pot odds describe the ratio of the pot size to the required amount of the call.  This odds ratio is then used to compare to the expected probability of a future card coming that will win the hand for you.  Pot odds are used when you have a drawing hand.  I also discuss pot odds as percentages even though they are technically odds ratios. 

Example 1

If you have an open ended straight-draw after the turn, then you have about a 17% chance of making the straight.  If the pot size is $100 and your opponent bets $20, should you call?  The odds of you making a straight to win the hand are 17% while the pot odds ratio is 16.67% (the $20 bet divided by the total pot size of $120…20/120).  Because the pot odds ratio of 16.67% is lower than the chances of you winning the hand on the river 17% of the time, you should make the call.  This can be demonstrated with a little bit of math.  If you make the call 100 out of 100 times, then you will theoretically lose 83% of the time.  On the 83 times you lose, it will cost you $20 every time to call totaling $1,660.  On the 17 times that you win, you will win $100 every time totaling $1,700.  Thus, making this call will net you a profit in the long run. 

Example 2

What if your opponent had bet $50 into the $100 pot?  The odds of you drawing a card that will make your straight are still 17%.  However, the pot odds ratio is now 33% (50/150).  This means that you should not make the call because the pot odds ratio is higher than the chances of you winning the hand.  Let’s take a look at the math.  On the 83 times that you lose, it will cost you $50 to make the call totaling $4,150.  On the 17 times that you win, you will win $100 every time totaling $1,700.  Thus, making this call will result in a net loss in the long run. 

Both of these scenarios are based on the assumptions that a made straight will be a stronger hand than your opponent’s hand and that there will be no future betting.  When you consider future betting, you are calculating implied pot odds.  Implied pot odds consider all of the money you can win if you make your hand.  Let’s take a look at another example.

Example 3

In the example above in which it costs $50 to call the bet, we found that you should not call because the pot odds are 33% while the chances of making the straight are 17%.  This is a perfect scenario to consider implied pot odds.  If you make the call and do not make your hand then you will lose no future money on betting because you can just fold.  If you make the call and do make your hand then you can bet with a winning hand.  If you bet $150 and then expect your opponent to call the $150 then the implied pot odds are 16.67% (50/(150+150) or 1:5).  Now your chances of making the straight are higher than the implied pot odds.  In fact, the numbers are the same as in example 1.  You will now make a profit by making this call every time in the long run. 

Gambler’s Fallacy is a fascinating concept.  Individuals cannot predict the outcome of any one independent event, and the probability of an independent event does not rely on past results.  However, the law of large numbers allows you to continue to make bets in situations such as when the pot odds are in your favor because if you play enough hands then you should make a profit in the long run regardless of your short run results.  I used to play poker a lot when I was in high school.  I made $25,000 over the course of two years using methods such as pot odds.  I have since quit playing poker, but the statistics behind the game will never cease to amaze me.

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