Birthday Month

Solution

Answer: 61.8

Explanation: Let each friend be equally likely to be born in any of the \(12\) months, independently of the others. We want the probability that at least two of the 5 friends share a birthday month. It is easier to compute the probability that all 5 friends have different birthday months and subtract from \(1\). For all 5 friends to have different birthday months: First friend: \( \frac{12}{12} \) Second friend: \( \frac{11}{12} \) Third friend: \( \frac{10}{12} \) Fourth friend: \( \frac{9}{12} \) Fifth friend: \( \frac{8}{12} \) Thus, \[ P(\text{all different}) = \frac{12}{12} \cdot \frac{11}{12} \cdot \frac{10}{12} \cdot \frac{9}{12} \cdot \frac{8}{12} \] \[ = \frac{11 \cdot 10 \cdot 9 \cdot 8}{12^4} \] \[ = \frac{7920}{20736} \] \[ \approx 0.3819 \] Now subtract from \(1\): \[ P(\text{at least one shared birthday month}) = 1 - P(\text{all different}) \] \[ = 1 - \frac{7920}{20736} \] \[ = \frac{12816}{20736} \] \[ \approx 0.6181 \] Therefore the probability that at least one pair shares a birthday month is \(0.6181 \approx 61.8\%\).