Question

Problem 2) Willy Wonka’s Chocolate Factory produces four types of chocolate bars: • 30 percent of the chocolate bars are milk chocolate of which 10 percent have a golden ticket, • 15 percent of the chocolate bars are bitter chocolate of which 20 percent have a golden ticket, • 35 percent of the chocolate bars are mint chocolate of which 40 percent have a golden ticket, • 20 percent of the chocolate bars are vegemite chocolate of which 5 percent have a golden ticket, (1 point) (i) What is the probability that a randomly selected chocolate bar is milk chocolate? (2 points) (ii) What is the probability that a randomly selected chocolate bar has a golden ticket? (2 points) (iii) If a randomly selected chocolate bar has a golden ticket, what is the probability that it is bitter chocolate?

Answer 1

We are given,

P(milk) = 30% = 0.30

P(Golden ticket | milk) = 10% = 0.10

P(bitter) = 15% = 0.15

P(Golden ticket | bitter) = 20% = 0.20

P(mint) = 35% = 0.35

P(Golden ticket | mint) = 40% = 0.40

P(vegemite) = 20% = 0.20

P(Golden ticket | vegemite) = 5% = 0.05

(i)

Probability that a randomly selected chocolate bar is milk chocolate = 30% = 0.30

(ii)

Probability that a randomly selected chocolate bar has a golden ticket = P(Golden ticket) =

= P(milk)*P(Golden ticket | milk) + P(bitter)*P(Golden ticket | bitter) + P(mint)*P(Golden ticket | mint) + P(vegemite)*P(Golden ticket | vegemite)

= 0.3*0.1 + 0.15*0.20 + 0.35*0.40 + 0.20*0.05 = 0.21

(iii)

P(bitter chocolate | Golden ticket) = P(bitter chocolate and golden ticket) / P(Golden ticket)

Now,

P(bitter chocolate and golden ticket) = P(bitter)*P(Golden ticket | bitter) = 0.15*0.20 = 0.03

P(Golden ticket) = 0.21

P(bitter chocolate | Golden ticket) = 0.03/0.21= 0.143