I heard a caller on a local radio station this evening say something in praise of her 17-year-old son. The host said, “You must have a good relationship with your son.”
“Well, I have three sons,” said the caller. “So I have a 33% chance of having a good relationship.”
Of course, this was a flippant remark not meant as a serious estimate of the probability of having a good relationship. But it got me to wondering about the issue.
Suppose relationships can be unequivocally classified as either good or bad, and that both are equally likely. Then the probability of a good relationship with one son is 50%. With three sons, the probability of at least one good relationship out of the three is 87.5%, much better than 33%. In fact, for the probability to be only 33% for at least one good relationship with three sons, the probability of a good relationship with one son would have to be about 12.5%.
Probability can be tricky.
Suppose you are on a jury in a murder trial. A key piece of evidence comes from an eyewitness who claims that he saw the accused getting into a yellow cab. In tests the witness reliably identifies yellow cabs as yellow 95% of the time. The other 5% he mistakes them for white. He also correctly identifies white cabs 80% of the time, but 20% of the time mistakes them for yellow. However, 90% of the city cabs are white, and 10% are yellow. What is the probability that the witness actually saw a yellow cab?
Suppose there are 1000 cabs in the city. 900 are white, and 100 are yellow. Of the 900 white cabs, the witness would correctly identify 720 as white and misidentify 180 as yellow. Likewise, he would correctly identify 95 of the yellow cabs as yellow and misidentify 5 as white. Therefore, he would identify 275 cabs as yellow of which 180 are really white. The probability that the cab was yellow is 95/275 = 35%. Despite the reliability of the witness, the actual number of cabs of each color makes the probability of a mistaken identification 65%. It could easily be enough for reasonable doubt.