Q: What is the probability that in a group of 31 people, none of them have birthdays in February or August?

Physicist: There are 365.25 days per year (on average), there are 31 days in August, and 28.25 days in February (on average).  The “.25″ isn’t exact, but the last time that the “leap year every four years” rule wasn’t used was 1900.

The chance that one person’s birthday is not in February or August is P = \frac{365.25 - 31 - 28.25}{365.25} = \frac{306}{365.25} = 83.78 \%.  The chance that all 31 people in a group don’t have birthdays in February and August (assuming there are no correlations between those people), is P^{31} = 0.41 \%.

All of this assumes that the probability of being born on any one day of the year is equal to any other, which is nearly true.

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