Conditional probability is the science of updating your probabilistic beliefs based on new information. For example, when rolling a fair, six-sided die, you believe the probability of rolling a 6 is \(\frac{1}{6}.\)

However, if a friend saw the number you rolled and told you it was even, then the conditional probability that you rolled a 6 (given that the roll was even) is \(\frac{1}{3}.\)

The conditional probability of “\(A\) given \(B\)” is denoted as \(P(A|B).\)

For two events \(A\) and \(B,\) is \(P(A|B)\) greater than, less than, or equal to \(P(A)?\)

You are on a game show in which there are 3 doors. Behind one door is a pot of gold, but the other doors contain rocks. You choose door #1.

According to the rules of the game, the host of the game will open one of the two doors you didn't choose and will reveal rocks (which he can guarantee, as he knows where the gold is). He opens door #3.

The host offers you a decision: stick with door #1, or switch to door #2. To maximize the probability that you get the gold, what should you do?

You are on a game show in which there are 3 doors. Behind one door is a pot of gold, but the other doors contain rocks. You choose door #1.

The host of the game show pulls a lever which will **randomly** open one of the two doors you didn't choose. To your relief, door #3 was opened and turned out to have rocks!

The host offers you a decision: stick with door #1, or switch to door #2. To maximize the probability that you get the gold, what should you do?

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