Logical Theology

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The Gambler’s Fallacy

When people gamble at a casino, they often make the mistake of assuming that some event is due to happen because of past events. Suppose that a coin is tossed 8 times, and all 8 times the coin lands tails. The probability of this occurring is 1/256 (one chance in 256 tosses). Ask yourself this question: if the coin is not fixed, is heads or tails more likely to come up?

The answer is that the odds of either heads or tails are 50/50.

The fact that the previous 8 tosses resulted in tails does not make it more likely that the next toss will be heads, assuming the coin is fair (having two different sides and equally weighted).  To think that heads is more likely to occur than tails is to commit the gambler’s fallacy.

In fact, if a coin is flipped 8 times, whatever combination of heads and tails comes up, the odds of that combination are still 1/256. It is true that if the coin is fair, the number of heads and tails will approach 50/50. However, the fact that heads came up 8 times in a row does not increase the odds of the next throw being tails. Thus, the gambler’s fallacy is committed when someone believes that after a number of lucky or unlucky events, the opposite outcome is more likely or due to occur. This overlooks the fact that there may be other factors which go into the situation. If an event is truly random, claiming that an unlikely event (good or bad, lucky or unlucky) will make a future event more likely is a fallacy.

Let’s consider the following examples:

Example 1:

Jack says, “I am going to win that 10 million-dollar lotto this month.”

Jill replies, “How do you know you will win?”

Jack responds, “Because I have been playing the lotto for twenty years straight.”

Jill replies, “You have been playing the lottery for over 20 years and have never won. Why would you be more likely to win this time?”

Jack responds, “Because after playing for so long, my odds of winning are much higher than they were in the past.”

What Jack fails to realize is that his odds of winning the lotto are exactly the same every time he plays. In essence, he believes that because he has been playing the lotto for twenty years, his chances of winning it this time are greater than those before. This is clearly not the case.  In reality, his chances of winning the lotto with a single ticket are just as small as they were every other time he played. The fact that he played for the last twenty years does not increase his chances in the present lottery.  Therefore, Jack has committed the gambler’s fallacy.

Gamblers commit this fallacy all the time. They assume that because the probabilities of something happening are low, they are going to place their bets against it happening. Thus, they assume that they are more likely to win a dice game because they have lost the previous six times. They might also assume that a roulette wheel will land on a different color because it has landed on red the previous dozen times. However, in thinking this, they have succumbed to the gambler’s fallacy. A throw of the dice is random. No matter how many times the dice are thrown, the fact that you lost six times does not increase the odds of having a favorable throw. No matter how many times the roulette wheel is spun, the odds of it landing on one color as opposed to the other are exactly the same for each spin. If it keeps landing on red, the logical conclusion is that the wheel is fixed, not that black is more likely.

Though the gambler’s fallacy gets its name from the gamblers who commit it, the gambler’s fallacy is not restricted to dice games and roulette. Many people in non-gambling situations can commit this fallacy as well.

Let’s consider another example.

Example 2:

Michael says, “I believe that the odds of the Mars rover landing on Mars are very good.”

Gloria says, “Why do you think that?”

Michael says, “Because we have lost the last 4 probes we sent there. Probability-wise, our odds are therefore greater that our mission will be successful.”

It may well be that the odds are better for this mission as opposed to the last one; however, this would be due to better science, technology, favorable conditions, and planning. While scientists can and do attempt to correct any errors and give their space missions the best chance of succeeding, if the odds of future Mars missions were increased, it would not be due to the fact that the last four missions were failures. The odds of success are not magically increased just because there have been numerous past failures. Unless technology improves, the odds for each mission’s success are exactly the same or nearly so. The fact that the last four probes were lost is not a factor in how likely it is that the current mission will succeed.

Consider one final example.

Example 3:

Jennifer says, “I do not think that you need to worry about purchasing that car.”

Florence responds, “Why not? I heard from five of my friends who bought past cars from the same company that their radiators overheated.”

Jennifer replies, “That is my point. After those failed car designs, the company is surely going to produce a good car.”

No, that is not necessarily the case. It may well be that the company has fixed the design flaws of their past car models. However, after manufacturing faulty car designs, Jennifer should insist on being given proof that the company’s current car design is not flawed.  Florence’s suggestion that because of past failures, the company is due to produce a good car design is an example of the gambler’s fallacy. The fact that the company has failed in the past does not automatically mean that are going to succeed now.

The Bottom Line: Believing that repeated lucky or unlucky events make the opposite event more likely to occur is a logical fallacy.

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