Understanding The Problem
The birthday problem asks whether any of the people in a given group has a birthday matching any of the others — not one in particular. (See "Same birthday as you" below for an analysis of this much less surprising alternative problem.)
In the example given earlier, a list of 23 people, comparing the birthday of the first person on the list to the others allows 22 chances for a matching birthday, the second person on the list to the others allows 21 chances for a matching birthday, third person has 20 chances, and so on. Hence total chances are: 22+21+20+....+1 = 253, so comparing every person to all of the others allows 253 distinct chances (combinations): in a group of 23 people there are pairs.
Presuming all birthdays are equally probable, the probability of a given birthday for a person chosen from the entire population at random is 1/365 (ignoring Leap Day, February 29). Although the pairings in a group of 23 people are not statistically equivalent to 253 pairs chosen independently, the birthday paradox becomes less surprising if a group is thought of in terms of the number of possible pairs, rather than as the number of individuals.
Read more about this topic: Birthday Problem
Famous quotes containing the word problem:
“What had really caused the women’s movement was the additional years of human life. At the turn of the century women’s life expectancy was forty-six; now it was nearly eighty. Our groping sense that we couldn’t live all those years in terms of motherhood alone was “the problem that had no name.” Realizing that it was not some freakish personal fault but our common problem as women had enabled us to take the first steps to change our lives.”
—Betty Friedan (20th century)