winter_born_girl.R

# Example 2.2.7 (A girl born in winter). Page 46.
# A family has two children. Find the probability that both children are girls, 
# given that at least one of the two is a girl who was born in winter. Assume 
# that the four seasons are equally likely and that gender is independent of season.
# Blitzstein, Joseph K.; Hwang, Jessica (2014-07-24). 
# Introduction to Probability (Chapman & Hall/CRC Texts in Statistical Science) 

library(stringr)
n <- 1000000

# P(GG | At least one Gw [Gw = Winter Girl])
# w: winter, s:spring, u: summer, f: fall
s <- c("Bw", "Bs", "Bu", "Bf", "Gw", "Gs", "Gu", "Gf")
ss <- as.character(outer(s, s, FUN = paste0))
pop <- sample(ss, n, replace = TRUE)

# P(At least one Gw), which
# should be close to 1 - (7/8)^2
sum(str_detect(pop, "Gw")) / n
1 - (7/8)^2

# P(GG and At least one Gw) 
# [`.` stands for any character, `|` means or]
# should be close to 1/4 * (1 - (3/4)^2)
sum(str_detect(pop, "GwG.|G.Gw")) / n
1/4 * (1 - (3/4)^2)

# P(GG | At leat one Gw) = P(GG, At least one Gw) / P(At least one Gw)
# should be close to 7/15
sum(str_detect(pop, "GwG.|G.Gw")) / sum(str_detect(pop, "Gw"))
7/15