Page 88 #4

Solution

This is similar to the birthday problem.

library(gmp)
## Warning: package 'gmp' was built under R version 4.2.3
## 
## Attaching package: 'gmp'
## The following objects are masked from 'package:base':
## 
##     %*%, apply, crossprod, matrix, tcrossprod
# five alive USA presidents
us_pres <- 5
year_day <- 365

# factorial of 365 to get a distinct day where a president pass away
a <- year_day - us_pres
b <- factorialZ(year_day)
c <- factorialZ(a)
d <- as.numeric(b/c)
e <- as.numeric(year_day ^ us_pres)

distinct_day <- d/e

distinct_day
## [1] 0.9728644

The probability of a distinct day is 97.28%

# probability of 2 presidents pass away on the same day
same_day <- 1 - distinct_day

same_day
## [1] 0.02713557

The probability that at least 2 US president pass away on the same day is 2.71% and that would be unfavorable for a bet.