Question

A game show offers contestants the following chance to win a car: There are three doors. A car is hidden behind one door, and goats are hidden behind each of the other doors. The contestant selects a door. The game show host then opens one of the doors not chosen to reveal a goat (there are two goats, so there is always such a door to open). At this point, the contestant is given the opportunity to stand pat (do nothing) or to choose the remaining door. Suppose you are the contestant, and suppose you prefer the expensive sports car over a not-so-expensive goat as your prize. What do you do?

(a) Suppose you decide to stand with your original choice. What are your chances of winning the car?

(b) Suppose you decide to switch to the remaining door. What are your chances of winning the car?

(c) Suppose you decide to flip a fair coin. If it comes up heads, you change your choice, otherwise, you stand pat. What are your chances of winning the car?

Answer 1

a)let you choose door 1 and host opens door 2

P(host opens door 2)

=P(prize behind door 1)*P(host opens door 2|prize behind door 1)+P(prize behind door 3)*P(host opens door 2|prize behind door 3)

=(1/3)*(1/2)+(1/3)*(1) =1/2

hence P(prize behind door 1 given host opens door 2)=P(chance if winnining the car with original choice)

=P(prize behind door 1)*P(host opens door 2|prize behind door 1)/P(host opens door 2)

=(1/3)*(1/2)/(1/2)=1/3

b)chances of winning the car by switching

=P(prize behind door 3)*P(host opens door 2|prize behind door 3)/P(host opens door 2)

=(1/3)*(1)/(1/2)=2/3

c)

P(chance of winnning )=P(heads)*P(change and win)+P(tails)*P(stay pat and wins)=(1/2)*(2/3)+(1/2)*(1/3)=1/2