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Conditional Probability

                 

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 \(P(A)?\)

Three fair, six-sided dice are rolled, and the sum of the numbers rolled is even. What is the probability that all three numbers rolled were even?

One of the most important theorems in conditional probability is Bayes’ Theorem. For two events \(A\) and \(B,\) \[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}.\]

On any given day, the probability that General Motors (GM) and General Electric (GE) both go up is 40% and the probability that they both go down is 20%. Otherwise, one goes up and the other goes down. If you know that at least one went up, what is the probability that both went up?

You have a jar with 4 fair coins and 1 unfair coin (which has heads on both sides). You choose one of the 5 coins at random, flip it five times, and get all heads. What is the probability that you chose the unfair coin?

Now, it's time to tackle the famous Monty Hall problem. The first problem will be the traditional problem -- one in which a relatively straightforward application of conditional probability gives an answer that befuddled several PhD mathematicians in 1990. The second (and final) problem will consider a slight variant.

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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