Monty Hall problem
Topic of Random experiments and probability, part of Probability
Monty Hall problem
| Type | Topic |
| Taught in | Random experiments and probability |
| Part | Probability |
| Worked examples | 1 |
A prize is behind one of three doors. You pick a door; the host, who knows where the prize is, opens one of the other doors to show no prize and offers a switch. Switching wins with probability 2/3, because it wins exactly when the first pick was wrong.
Examples
The Monty Hall problem
Worked example from Random experiments and probability
A prize is equally likely behind curtain A, B or C. The contestant picks A. The host, who knows where the prize is, always opens a different curtain that has no prize, choosing at random when both remaining curtains are empty.
Question. Suppose the host opens B. Should the contestant switch to C?
Solution. The definition of conditional probability settles it without any story about psychology. The four atoms that can occur, with their probabilities, are: prize behind A and host opens B (1/6), prize behind A and host opens C (1/6), prize behind B and host opens C (1/3), prize behind C and host opens B (1/3). The observed event is the two atoms in which B is opened, with total probability 1/6+1/3=1/2. The prize is behind A in only the first of those atoms, so
\mathbb{P}(\text{prize behind }A\mid \text{host opens }B)=\frac{1/6}{1/2}=\frac13,
and the probability it is behind C is 2/3. Switching wins with probability 2/3.
The tempting answer, one half each, forgets that the host’s choice depends on where the prize is. When the contestant’s first pick is wrong, which happens two times in three, the host is forced to reveal the only other empty curtain.
Used in
See also
Sample space · Event · Probability measure · Properties of a probability · Equally likely outcomes · Four ways of counting draws · Conditional probability · Product rule