So far we have learned how to calculate the probability of a single event happening, but sometimes life isn't so simple as one event happening by itself; sometimes we're interested in two events happening together, or one after another. Calculating the probability of two events both happening isn't terribly complicated; we just calculate the two individual probabilities, and multiply them together. We do need to pay attention to whether the events are dependent or independent; if the events are dependent, our sample space (possible outcomes) may change, and our number of desired outcomes may change, based on the first event.

Example One
If I roll a four-sided die, and a six-sided die, what is the probability that they will both show the number one?

Answer
These are independent events. First we calculate the probability of getting a one on the four-sided die. That's . Now we calculate the probability of getting a one on the six-sided die. That's .

Thus, the probability of ones showing up on both dice is x = .

This answer makes a lot of sense, because we could have calculated it a different way:

If you consider the two die rolls as a single event, there are 24 possible outcomes (see our fundamental counting principle page for an explanation), and only one of them is desired.

Example Two
A jar has 20 red jelly beans and 5 green ones. I pick two jelly beans. What is the probability that they are both green?

Answer
 To solve this, we treat it as two events. First I pick a jelly bean, then I pick another one. These are dependent events, because the first event affects the second one.

First jelly bean: there are 25 possible outcomes, and 5 of them are desired: = .

Second jelly bean: there are now 24 possible outcomes (I've taken one jelly bean!), and 4 of them are desired (because I've already taken a green one!). = .

Thus, the probability of getting both greens is x = .

Example Three
My sock drawer has 5 red socks and 4 blue socks. If I randomly pick two socks, what is the probability that I'll have a pair of blue socks?

Answer
First sock: 9 possible, 4 desired. P =

Second sock: 8 possible, 3 desired. P =

Probability of a blue pair: x =

1.

What is the probability that I will get heads twice, if I flip a fair coin twice?

2.

What is the probability that I will get a head, followed by a tail, if I flip a fair coin twice?

3.

I have ten marbles in a jar. Three are red, four our black, two are green, and one is yellow. If I draw two marbles, what is the probability that they are both red?

4.

I draw two cards from the deck. What is the probability that the first card is red, and the second card is black?

5.

I draw two cards from the deck. What is the probability that both cards are red?

6.

In the previous two questions, the two answers are not the same. Can you explain why not?