Birthday problem

In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday. P ( A ′ ) = 365 365 × 364 365 × 363 365 × 362 365 × ⋯ × 343 365 {displaystyle P(A')={frac {365}{365}} imes {frac {364}{365}} imes {frac {363}{365}} imes {frac {362}{365}} imes cdots imes {frac {343}{365}}}     (1) P ( A ′ ) = ( 1 365 ) 23 × ( 365 × 364 × 363 × ⋯ × 343 ) {displaystyle P(A')=left({frac {1}{365}} ight)^{23} imes (365 imes 364 imes 363 imes cdots imes 343)}     (2)The reasoning is based on important tools that all students of mathematics should have ready access to. The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the multiplications would take much longer, and be much more subject to error, whether the instrument is a pencil or an old-fashioned desk computer. What calculators do not yield is understanding, or mathematical facility, or a solid basis for more advanced, generalized theories. In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday. Actual birth records show that different numbers of people are born on different days. In this case, it can be shown that the number of people required to reach the 50 percent threshold is 23 or fewer. For example, if half the people were born on one day and the other half on another day, then any two people would have a 50 percent chance of sharing a birthday.