The general multiplication rule

When we calculate probabilities involving one event AND another event occurring, we multiply their probabilities.

In some cases, the first event happening impacts the probability of the second event. We call these dependent events.

In other cases, the first event happening does not impact the probability of the seconds. We call these independent events.

What is the probability of flipping a fair coin and getting "heads" twice in a row? That is, what is the probability of getting heads on the first flip AND heads on the second flip?

Imagine we had $100$‍ people simulate this and flip a coin twice. On average, $50$‍ people would get heads on the first flip, and then $25$‍ of them would get heads again. So $25$‍ out of the original $100$‍ people — or $1/4$‍ of them — would get heads twice in a row.

The number of people we start with doesn't really matter. Theoretically, $1/2$‍ of the original group will get heads, and $1/2$‍ of that group will get heads again. To find a fraction of a fraction, we multiply.

We can represent this concept with a tree diagram like the one shown below.

We multiply the probabilities along the branches to find the overall probability of one event AND the next even occurring.

For example, the probability of getting two "tails" in a row would be:

When two events are independent, we can say that

Suppose that we are going to roll two fair $6$‍-sided dice.

We can use a similar strategy even when we are dealing with dependent events.

Consider drawing two cards, without replacement, from a standard deck of $52$‍ cards. That means we are drawing the first card, leaving it out, and then drawing the second card.

What is the probability that both cards selected are black?

Half of the $52$‍ cards are black, so the probability that the first card is black is $26/52$‍. But the probability of getting a black card changes on the next draw, since the number of black cards and the total number of cards have both been decreased by $1$‍.

Here's what the probabilities would look like in a tree diagram:

So the probability that both cards are black is:

A table of $5$‍ students has $3$‍ seniors and $2$‍ juniors. The teacher is going to pick $2$‍ students at random from this group to present homework solutions.

The vertical bar in $P(\text{B}|\text{A})$‍ means "given," so this could also be read as "the probability that B occurs given that A has occurred."

This formula says that we can multiply the probabilities of two events, but we need to take the first event into account when considering the probability of the second event.

If the events are independent, one happening doesn't impact the probability of the other, and in that case, $P(\text{B}|\text{A})=P(\text{B})$‍.