"Writing around 1564, Girolamo Cardano refers to a certain unlikely dice throw and assures us that 'still in an infinite number of throws, it is almost necessary for it to happen.' Cardano is emphasizing that with many trials the number of times a particular outcome will occur is very close to mathematical conjecture, or mathematical expectation. And this applies to even the most unlikely events; if it is possible for them to happen, given enough opportunities, eventually they will happen, in accordance with the laws of probability."

Here are a few of those seemingly extraordinary events that occasionally happen, when given enough opportunities.

--page 76, Randomness

"The headline of a 1990 New York Times article reads as follows: '1-in-a-Trillion Coincidence, You Say? Not Really, Experts Find.'¹ The article goes on to report a seemingly unbelievable coincidence about a woman who won the New Jersey lottery twice within four months, a feat originally reported as a 1 in 17 trillion long shot. Research on coincidences by two Harvard statisticians revealed, however, that the odds of such an event happening to someone somewhere in the United States were more like 1 in 30--not that amazing after all. They explain that this is an example of the law of very large numbers: 'With a large enough sample, any outrageous thing is likely to happen.'² Out of the millions upon millions of people who regularly purchase lottery tickets in the United States, it is not unreasonable that someone should at some point hit the lottery twice."

--page 72, Randomness
1. Kolata, 1990
2. Diaconis and Mosteller, 1989 (p. 859).

"Another article in the New York Times...regarding the tragic TWA Flight 800 crash...reported that 'more than once, senior crash investigators have tried to end the speculation by ranking the possibility of friendly fire at about the same level as that a meteorite destroyed the jet.' In a letter to the editor..., Charles Hailey and David Helfand wrote: 'The odds of a meteor striking TWA Flight 800 or any other single airline flight are indeed small. However, the relevant calculation is not the likelihood of any particular aircraft being hit, but the probability that one commercial airliner over the last 30 years of high-volume air travel would be struck by an incoming meteor with sufficient energy to cripple the plane or cause an explosion.' Noting that 3000 meteors with the requisite mass hit the earth every day, and that 50,000 commercial airline flights occur worldwide each day; and assuming an average flight time of two hours, which translates into more than 3500 planes in the air at any moment; and calculating that these planes would cover approximately two-billionths of the earth's surface, the authors conclude that, in over 30 years of air travel, the probability that a commercial flight would have been hit by meteoric impact sufficient to crash the plane is 1 in 10."

--page 73, Randomness

"In her weekly column in Parade Magazine, the enormously popular Marilyn vos Savant, purported by the Guinness Book of World Records to have the world's highest IQ, is asked the following question by a reader, 'Somehow you've overcome extreme odds and flipped 100 consecutive heads. The chances of flipping another head on the next toss can't possibly be as great as 50-50, can they?'³ Indeed, many people feel as this reader does, that 100 heads is sure to be soon offset by a tail. But as the reader herself points out, a person who flips 100 heads in a row has already overcome extreme odds to get where she is. The question now is only about one toss. If the coin is fair, her chances are indeed 50-50, the same as they were on the first toss. But perhaps the 100 previous tosses present some evidence that the coin is not fair--it might, in fact, be biased toward heads. In that case, the chances of a head on the next toss are better than 50-50."

--page 79-80, Randomness
3. Vos Savant, December 26, 1993 (p. 3), quoted in CHANCE website.

Return to RANDOMNESS