Discussion 5

library(gmp)

Exercise 18

A baker blends 600 raisins and 400 chocolate chips into a dough mix and, from this, makes 500 cookies. (a) Find the probability that a randomly picked cookie will have no raisins. (b) Find the probability that a randomly picked cookie will have exactly two chocolate chips. (c) Find the probability that a randomly chosen cookie will have at least two bits (raisins or chips) in it.

#Lets use a function to calculate combinations using gmp
choose_gmp <- function(n, k) {
  as.bigq(factorialZ(n) / (factorialZ(k) * factorialZ(n - k)))
}

# (a) Probability of no raisins
prob_no_raisins <- as.numeric(choose_gmp(400, 0) * choose_gmp(600, 500) / choose_gmp(1000, 500))

# (b) Probability of exactly 2 chocolate chips
prob_2_chips <- as.numeric(choose_gmp(400, 2) * choose_gmp(600, 498) / choose_gmp(1000, 500))

# (c) Probability of at least two bits
prob_at_least_two_bits <- 1 - prob_no_raisins - as.numeric(choose_gmp(400, 1) * choose_gmp(600, 499) / choose_gmp(1000, 500))

#Displaying the results
prob_no_raisins

## [1] 4.111948e-184

prob_2_chips

## [1] 7.946932e-178

prob_at_least_two_bits

## [1] 1