Question

Problem 19.34.

Peeta bakes between 1 and 2n loaves of bread to sell every day. Each day he rolls

a fair, n-sided die to get a number from 1 to n, then flips a fair coin. If the coin is

heads, he bakes m loaves of bread , where m is the number on the die that day, and

if the coin is tails, he bakes 2m loaves.

(a) For any positive integer k <= 2n, what is the probability that Peeta will make

k loaves of bread on any given day?

Hint: Express your solution by cases.

(b) What is the expected number of loaves that Peeta would bake on any given

day?

(c) Continuing this process, Peeta bakes bread every day for 30 days. What is the

expected total number of loaves that Peeta would bake?

Answer 1

Probability of getting all the numbers on die is equal = 1/n

a) Exactly k loaves of bread any day

P[ getting k on die ] = 1/n , and P[ Head on coin ] = 1/2

P[ getting k/2 on die ] = 1/n, and P[ Tail on coin ] = 1/2

P[ Baking exactly l breads ] = P[ getting k on die and getting head on coin ] + P[ getting k/2 on die and getting tail on coin ]

Since, the events are independent

P[ Baking exactly l breads ] = P[ getting k on die ]*P[ Head on coin ] + P[ getting k/2 on die ]*P[ Tail on coin ] = ( 1/2n ) + ( 1/2n ) = 1/n

b) Expected number of breads any day = summation(m = 1 to n )( m*p(m)) = summation(m = 1 to n )( m/n) ( Since p(m) , baking m number of breads = 1/n )

Expected number of breads any day =summation(m = 1to n )( m ) / n = (n(n+1)/2) / n = n(n+1)/2n = (n+1)/2

c) Expected number of total breads he bake in 30 days = 30*Expected number pf breads any day = 30*(n+1)/2 = 15(n+1)