Question

Consider the game of 'Let's Make a Deal' in which there are three doors (numbered $1,2,3$), one of which has a car behind it and two of which are empty (have "booby prizes"). You initially select Door $1$, then, before it is opened, Monty Hall tells you that Door $3$ is empty (has a booby prize). You are then given the option to switch your selection from Door $1$ to the unopened Door $2$. What is the probability that you will win the car if you switch your door selection to Door $2$?

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. None of these

Answer 1

Answer: (C)

$S=\{1,2,3\}$, where outcome ${}^{\prime }{i}^{\prime }$ means that the car is behind door $i$. Let $E=$ {Door $3$ is empty} $=\{1,2\}$. The probability that you win by switching to Door $2$, given that he tells you Door $3$ is empty is
$P\left(\left\{2\right\}|\left\{1,2\right\}\right)=\frac{P\left(\left\{2\right\}\cap \left\{1,2\right\}\right)}{P\left(\left\{1,2\right\}\right)}$
$=\frac{1/3}{2/3}=\frac{1}{2}$