In this short clip I describe the Monty Hall problem, and suggest a way of understanding its somewhat counter-intuitive solution. The Monty Hall problem can be described described elegantly using Bayes’s theorem.
Recall that:
<br />
P(A | B) P(B) = P(A, B)<br />
P(B | A) P(A) = P(A, B)<br />
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Combining these two statements yields Bayes’s theorem:
<br />
P(A | B) = P(B | A) P(A) / P(B)<br />
Bayes developed this theorem as part of his quest to prove the existence of God. For example, what is P(god | what we observe) is a hard problem, but a more tractable problem may be: what is P(what we observe | god) P(god). In other words, this theorem is extremely useful because it allows us to “flip around” many questions in statistics.