Introduction to Cryptography with Coding Theory, 3rd Edition

C.8 Examples for Chapter 12

Example 36

Suppose there are 23 people in a room. What is the probability that at least two have the same birthday?

SOLUTION

The probability that no two have the same birthday is $\prod _{i=1}^{22}(1-i/365)$ (note that the product stops at $i=22$, not $i=23$). Subtracting from 1 gives the probability that at least two have the same birthday:

>> 1-prod( 1 - (1:22)/365)

ans =
   0.5073

Example 37

Suppose a lazy phone company employee assigns telephone numbers by choosing random seven-digit numbers. In a town with 10,000 phones, what is the probability that two people receive the same number?

>> 1-prod( 1 - (1:9999)/10^7)

ans =
   0.9933

Note that the number of phones is about three times the square root of the number of possibilities. ...