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Notes on Probability: Formula for Conditional Probabilities

I'm reading Peter Olofsson's book, "Probabilities: The Little Numbers That Rule Our Lives". Here is the formula for conditional probabilities:

Given events A and B,

P(A and B) = P(A) × P(B given A)

In other words, the probability of events A and B both happening are the same as the probability of event A happening, multiplied by the probability of B happening if you known A has already happened or is certain to happen.

If the events A and B are independent (one event happening does not affect the likelyhood of the other happening) then the formula reduces to

P(A and B) = P(A) × P(B)

A simple example of the conditional probability formula: Suppose you are flipping a coin twice. Let's say that A is the event that at least one of the coins will land on heads, and B is the event that at least one of the coins will land on tails. What then is P(A and B), the probability that you will get at least one coin landing on heads and one coin landing on tails?

We know there are four possible outcomes of the coin flipping:

HH
HT
TH
TT

Three out of those four possiblities have at least one heads, so the P(A) = 3/4. Now, what is P(B given A)? Since we must assume at least one heads, there are only three possible outcomes: HH, HT, TH. Two of these possibilities have at least one tails. So, P(B given A) = 2/3.

So, P(A and B) = P(A) × P(B given A) = 3/4 × 2/3 = 1/2, or 50% chance. Since this is such a simple example, you can verify that simply by looking at the four possible outcomes, and seeing that only two out of four of them have at least one heads coin and one tails coin.

For a more practical example: What is the probabity of drawing two cards out of a 52 card deck, and both of them being aces? Let event A be the drawing an ace as the first card, and event B be drawing an ace as the second card. There are four aces in the deck, so P(A) = 4/52. What about P(B given A)? If we have drawn one ace, there are now 3 aces in a 51 card deck, so P(B given A) = 3/51. So, P(A and B) = 4/52 × 3/51 = 1/221, which is to say the odds are 220 to 1.

Now, keep in mind, the odds are slightly better for the guy you are playing with, if you've got two kings, and are worried about him getting two aces. I'll leave that as an exercise for the reader.