There are about 7.6 Bn people on earth. 7,600,000,000 people! Understanding the sheer number of people out there in the world can help you make sense of the commonness of coincidences or improbable events. You should not be surprised by coincidences. In fact, you should expect coincidences and miracles to happen.
The New York Times ran a story about a woman who won the New Jersey lottery twice. “What a coincidence,” you’d say! It happens only 1 in 17 trillion times. How? Well, if you were to buy just two lottery tickets—one for a lottery where you need 6 correct numbers out of 39 to win; the other where you need 6 out of 42—these are the two lotteries that the woman from New Jersey won—the chance that these two tickets would both be winners is 1 in 17 trillion. Extreme luck!
However, statisticians Stephen Samuels and George McCabe of Purdue University begged to differ. “We’d be happy to bet even money that there will be another double winner somewhere in the US sometime in the next seven years,” they wrote. “Give us 2-to-1 odds, and we’ll shorten that to four years; make it 30-to-1, and we’ll bet on four months, which happens to be the time between winnings by the New Jersey woman.”
Why? Because players don’t buy one ticket for each of two lotteries. They buy multiple tickets from multiple lotteries every week. With so many people buying so many lotteries every week, somebody is bound to get lucky more than once.
The flip side of that event happened in the Bulgarian lottery. They randomly selected the winning numbers 4, 15, 23, 24, 35, 42 on September 6, 2009. Four days later it selected the same numbers again. It was shocking. Bulgaria’s then sports minister Svilen Neikov even ordered an investigation.
This rather stunning coincidence was simply an example of the law of truly large numbers. The idea here is that with a large enough group, or sample, unlikely things are completely likely to happen.
First, many lotteries are conducted around the world regularly rolling out their numbers. Second, they occur time after time, year in and year out. This rapidly adds up to a large number of opportunities for lottery numbers to repeat. And third, each time a lottery result is drawn, it could contain the same numbers as produced in any of the previous draws. “If you consider the number of lotto drawings—every week—around the world—over years and years—it would be amazing he wrote, if draws did not occasionally repeat.”
In fact, the North Carolina Cash 5 lottery produced the same winning numbers on July 9 and July 11, 2007. Strange? Not according to probability.
For a simplified example of the law, assume that a given event happens with a probability for its occurrence of 0.1%, within a single trial. Then, the probability that this so-called unlikely event does not happen (improbability) in a single trial is 99.9% (0.999).
In a sample of 1000 independent trials, however, the probability that the event does not happen in any of them, even once (improbability), is 0.999¹⁰⁰⁰, or approximately 36.8%. Then, the probability that the event does happen, at least once, in 1000 trials is 1 − 0.999¹⁰⁰⁰ ≈ 0.632 or 63.2%. This means that this “unlikely event” has a probability of 63.2% of happening if 1000 independent trials are conducted.
This gets even more interesting. The probability that it happens at least once in 10,000 trials is 1 − 0.999¹⁰⁰⁰⁰ ≈ 0.99995 = 99.995%. In other words, a highly unlikely event, given enough trials with some fixed number of draws per trial, is even more likely to occur.
Law of large numbers can also help you explain the wild success of some entrepreneurs.
“I am not saying that Warren Buffett is not skilled; only that a large population of random investors will almost necessarily produce someone with his track records just by luck.”
— Nassim Nicholas Taleb, Fooled by Randomness
99 out of 100 startups fail, but with a large number of startups out there, a lot of them are bound to enjoy stupid success.