Chance: A Guide to Gambling, Love, The Stock Market, and Just About Everything Else
By Amir D. Aczel (1950-2015)
This is a book about probability, the “quantitative measure of the likelihood of a given event.” The author applies probability theory in numerous scenarios.
Assuming a World War II pilot had a 2% chance of being shot down on each mission, what are the chances of a pilot being shot down in 50 missions? Nope—it is not 50 x 0.02. Using the law of unions of independent events, the answer is 1 – 0.9850 = 64%. In another example, there are three overnight couriers with an on-time record of 90%, 88%, and 92% respectively. If someone sent an important document using all three services, what is the probability of at least one of them delivering on time? The answer is 1 – (0.10 x 0.12 x 0.08) = 99.904%.
“So what have we noticed here? We’ve noticed that as the number of trials goes up, so does the probability of success… Suppose that my probability of getting a job is a mere half a percent, that is, 0.005. My claim is that if I can just persevere and apply at a very large number of firms, I can bring my probability of getting at least one job offer to a virtual certainty as well… If I apply for two thousand jobs, my probability of getting at least one job offer is: 1 – 0.9952000 = 99.9956%.”
“This also helps explain the increase in prostate cancer in men in America; as they are living longer and not dying from other causes, their odds of developing the otherwise uncommon cancer increase to 50%.”
However, perseverance doesn’t always guarantee success, such as with gambling. “The reason for this is that trials aren’t free—you have to pay for each gamble.” Another factor in gambling is that the games are designed to give the house a better chance of winning. “But let’s suppose that the game is fair… A famous mathematical theorem states that even in such a fair game, when you play against a much wealthier adversary, like a casino, given enough [bets], the gambler will lose with probability 1—or absolute certainty. This is called the Gambler’s Ruin Theorem.”
“Pascal’s triangle gives us a way of computing the probabilities of the number of heads or tails in any number of coin tosses (or the number of boys and girls in any number of children born to a couple, or any of similar equal-probability phenomena) … If the chances of a boy and a girl are equal, then by Pascal’s triangle only one of every 32 families with five children will have all girls or all boys… Notice an interesting property of probabilities that is apparent from inspecting the numbers in Pascal’s magic triangle: as the number of trials (tosses of the coin) increases, the probability of an even split becomes smaller! … Try this with larger numbers of tosses. If you toss a coin 12 times, the probability of an even split of heads and tails is as small as 23%.”
“The inspection paradox is one of my favorite topics in probability theory… Mathematically, what the inspection paradox says is that a probability distribution of a quantity that has already started its life is shifted, leading to a larger average than would otherwise be expected… Immigrants skew the longevity statistic upward. The reason is simple: an immigrant, arriving at his or her new country at a certain age, can no longer die at any age younger than the present. Let’s look at an extreme case. Suppose you have a country where everyone is an immigrant, and everyone arrives in this country at age 80. Clearly, such a country will have an inflated longevity—some age higher than 80.”
In another example, the light bulb you have in your lamp has a longer life than average. “Before an inspection, meaning before the light bulb is used, the average total lifetime of the light bulb is some number—in this case, two thousand hours. But let’s think a minute: this stated average incorporates the probabilities that the light bulb will fail in its first hour of operation (plus its first week, its first month, etc.). But your light bulb—the one that’s in right now—has already endured, surviving its first day, first week, first month. It can no longer fail in any earlier time than the present; therefore its total expected lifetime is longer than that of a light bulb you just bought… The same holds for batteries—and people’s lives. A person alive today will live, on average, longer than the expected longevity for his or her gender or nationality or ethnicity or any other category… For example, someone alive today can no longer die from infant mortality.”
The author also explains why you should always expect to have a longer than average wait at the bus stop.
With 23 people in the room, there is a 50% chance that two people will share a birthday. “Even more surprising, when 56 people are present in a room, there is a 99% probability that at least two of them share a birthday… In nature, we find much more aggregation—due to pure randomness, rather than despite of it—than we might otherwise expect… The birthday problem with all its attendant expansions is a problem of aggregated coincidences.”
“If, for example, you wanted to know what the probability is that in a group of 23 people, including yourself, at least one person will match your birthday.” We go back to the probability of the union of independent events: 1 – (364/365)22 = 1-0.94 = 6%. “And with 365 people in the room, that probability is still only 63%.”
Mathematically, what’s the best strategy to find the optimal spouse? “Suppose over a lifetime you expect to meet one hundred available candidates. If you marry the first one, the chance that you have indeed found the best of all one hundred candidates is only 1/100. Likewise, if you wait to meet all one hundred candidates, you will have rejected the 99 who came before, and the possibility that the last person you meet is also the best is again only 1/100. The best strategies allow you to sample for a while, in order to learn about the various candidates; and of all such strategies, the best has you sampling 37% of the total and then choosing the first candidate thereafter who beats all the ones who came before. Of course, there’s a chance you will never find one who is better than all 37% you’ve already seen.”
Aczel, Amir D. Chance: A Guide to Gambling, Love, the Stock Market & Just about Everything Else. New York: Thunder’s Mouth Press, 2004. Buy from Amazon.com
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