A wise man once said, “Never tell me the odds”, but whether you calculate the odds of successfully navigating an asteroid field (3,720:1), shouting “Shazam” and making it work twice in a row (9 million: 1), or winning the state lottery (42 million: 1 in California), odds influence the outcomes in our daily lives for events large and small. But given the widespread role they play in our lives, the average person is usually pretty good at calculating them accurately. As we see in the excerpt below from James C. Zimring’s latest title, Partial Truths: How Fractions Distort Our Thinking, our expectations regarding the probability of an event can shift depending on how the question is asked and at what fraction. is focused.
Columbia University Press
Excerpt from Partial Truths: How Fractions Distort Our Thinking by James C. Zimring, published by Columbia Business School Publishing. Copyright (c) 2022 James C. Zimring. Used in consultation with the publisher. All rights reserved.
Confusing the probable with the seemingly impossible: miscalculating the numerator
The more improbable an event seems, the more it grabs our attention when it occurs and the more we feel compelled to explain why it happened. This just makes sense. If the world doesn’t behave according to the rules we understand, we may be misunderstanding the rules. Our attention must be drawn to improbable events, because new knowledge arises from our attempts to understand contradictions.
Sometimes what seems impossible is in reality highly probable. A famous example of this can be found when playing the lottery (ie the lottery fallacy). It is common knowledge that it is incredibly unlikely that any particular person will win the lottery. For example, the probability of a particular ticket winning the Powerball lottery (the specific lottery analyzed in this chapter) is 1/292,000,000. This explains why so much attention is paid to the winners. Where did they buy their ticket? Have they seen a fortune teller before buying their ticket, or have they shown psychic abilities in the past? Do they have any special rituals they perform before buying a ticket? It is natural to try to explain how such an unlikely event could have happened. If we can identify a reason, perhaps understanding it can also help us win the lottery.
The lottery fallacy is not limited to good things that happen. Explanations are also sought to explain bad things. Some people are struck by lightning more than once, which just seems too unlikely to be a random coincidence. There must be an explanation. There is inevitably speculation that the person has some strange mutated property that makes them attract electricity, or they carry certain metals or have titanium prosthetics in their bodies. Perhaps they have been cursed by some mystical power or God has forsaken them.
The lottery fallacy can be understood as a form of confusing one chance for another, or continuing our theme from Part 1, confusing one fraction for another. One can express the probability of winning the lottery as the fraction (1/292 million), where the numerator is the single number combination that wins and the denominator is all possible number combinations. The misconception arises because we tend to notice only the one person with that one ticket who won the lottery. However, this is not the only person who plays the lottery, and it is not the only lottery. How many tickets are purchased for a particular draw? The exact number changes, as more tickets are sold the higher the jackpot; however, a typical drawing contains about 300 million tickets sold. Of course, some of the tickets sold must be duplicates, as there are only 292 million possible combinations. In addition, if every possible combination were bought, someone would win every draw. In reality, about 50 percent of the drawings have a winner; so we can deduce that on average 146 million different number combinations are bought.
Of course, the news doesn’t give us a list of all the people who didn’t win. Can you imagine the same headline every week, “299,999,999 people didn’t win the lottery again!” (names listed online at www.thisweekslosers.com). No, the news only tells us that there was a winner, and sometimes who the winner was. If we ask ourselves, “What is the probability that that person will win?” we are asking the wrong question and referring to the wrong fraction. The probability of that person winning is 1/292,000,000. Just by chance, that person should win the lottery once every 2,807,692 years they play consistently (assuming two draws per week). What we should be asking is, “What are the odds of someone winning?”
In probability, the probability of one or the other happening is the sum of the individual probabilities. So assuming there are no duplicate tickets, if only one person were to play the lottery, then the probability of a winner is 1/292,000,000. If two people are playing, the probability of a winner is 2/292 million. If 1,000 people are playing, the odds are 1,000/292,000,000. If we consider that 146 million different number combinations have been bought, the top of the fraction (numerator) becomes incredibly large and the probability that someone will win is quite high. When we are surprised that someone has won the lottery, we confuse the real fraction (146 million/292,000,000) for the fraction (1/292 million) – that is, we are misjudging the numerator. What seems an incredibly unlikely event is actually very likely. The human tendency to make this mistake is related to the availability heuristic, as described in Chapter 2. Only the winner is “available” to our minds, not all the many people who have not won.
Likewise, the chance of being struck by lightning twice in your lifetime is one in nine million. Since there are 7.9 billion people on Earth, there is a good chance that 833 people will be struck by lightning (at least) twice in their lifetime. As with the lottery example, our attention is only drawn to those who are struck by lightning. We don’t think about how many people never get spanked. Just as it is unlikely that any particular person will win the Powerball lottery, it is highly unlikely that anyone will win the lottery after a few draws given the number of people involved. Likewise, it is highly unlikely that a person will be struck by lightning twice, but it is even more unlikely that anyone will, given the number of people in the world.
So when we’re puzzling over such amazing things as someone winning the lottery or being struck by lightning twice, we’re actually trying to explain why something most likely happened, which really needs no explanation at all. The rules of the world work exactly as we understand them, but we confuse the highly probable for the nearly impossible.
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