What is the probability of dying on your birthday?
The cycle of leap year repeats every four years. In this four year period there are
\[364 \times 3 + 365 \times 1\]
days. (every four years an extra day, February 29th is added).There are therefore
\[364 \times 4 +1\]
ways to die on your birthday in a four year period.
There are \[9 \times 364 \times 364 + 6 \times 364 \times 365 + 1 \times 365 \times 365\]
combinations of birthdays and death days.The first term here represents birth and death days in none leap years. Three out of four years are non leap years so there are three times three combinations.
The second term represents birthday in a non leap year and death in a leap year or vice versa. There are three times one plus one time three ways.
The third term represents birth and death in a leap year.
The probability of dying on your birthday is then
\[\frac{364 \times 3 + 365 \times 1}{9 \times 364 \times 364 + 6 \times 364 \times 365 + 1 \times 365 \times 365}\]