There have been 44 presidents of the United States. Of those presidents, 38 are dead. After looking at the Wikipedia page for the List of Presidents of the United States by Date of Death, I decided to expand the book’s question to calculate the probability that at least two presidents have died on the exact same day in history (and not just the same day of the year).
Two presidents have achieved this distinction. Thomas Jefferson and John Adams both passed away on July 4, 1826. Therefore, the real number of presidents who died on the exact same day is \(\frac{2}{38} \approx\) 5.3%.
George Washington took office on April 30, 1789. The last president to have died, Gerald Ford, passed away on December 26, 2006. We’ll make the (very big) assumption that each president has an equal chance of dying on any day between those two points.
This means there are 79,259 possible days that U.S. presidents could have died. How likely is it that two presidents died on the same day?
years <- 2006-1789
# Taking leap years into account
leaps <- round(years/4)
days <- (years * 365) + leaps
days## [1] 79259With 38 dead presidents, we have 703 possible ways to pair them up:
\[ \frac{38 \times 37}{2} = 703 \text{ possible pairs} \]
Thus, the chance of taking 703 pairs with 79,259 possible outcomes each, and having at least one match, is tiny — just 0.9%.
\[ 1 - (\frac{79 \ 258}{79 \ 259})^{703} \approx 0.009 = 0.9\text{%} \]
I would not place a bet that any presidents died on the exact same day, since the chance that it has happened is highly unlikely. This makes it interesting that it happened at all.