A deck of ordinary cards is shuffled and 13 cards are dealt. What is the probability that the last card dealt is an ace?
#if there are no aces in first 12
p1 <- choose(48,12)*choose(4,0)/choose(52,12)*choose(4,1)/choose(40,1)
#if there is 1 ace in first 12
p2 <- choose(48,11)*choose(4,1)/choose(52,12)*choose(3,1)/choose(40,1)
#if there are 2 aces in first 12
p3 <- choose(48,10)*choose(4,2)/choose(52,12)*choose(2,1)/choose(40,1)
#if there are 3 aces in first 12
p4 <- choose(48,9)*choose(4,3)/choose(52,12)*choose(1,1)/choose(40,1)
#if there are 4 aces in first 12 (probability equal to 0)
p5 <- choose(48,8)*choose(4,4)/choose(52,12)*choose(0,1)/choose(40,1)
total_probability <- p1+p2+p3+p4+p5
total_probability## [1] 0.07692308