- Know the definition of conditional probability.
- Use conditional probability to solve for probabilities in finite sample spaces.
Tree diagrams are another way to show the outcomes of simple probability events. In a tree diagram, each outcome is represented as a branch on a tree.
Let’s say you were going to toss a coin 2 times and wanted to find the probability of getting 2 heads. This is an example of independent events, because the outcome of one event does not affect the outcome of the second event. What does this mean? Well, when you flip the coin once, you have an equal chance of getting a head (H) or a tail (T). On the second flip, you also have an equal chance of getting a a head or a tail. In other words, whether the first flip was heads or tails, the second flip could just as likely be heads as tails. You can represent the outcomes of these events on a tree diagram.
Irvin opens up his sock drawer to get a pair socks to wear to school. He looks in the sock drawer and sees 4 red socks, 8 white socks, and 6 brown socks. Irvin reaches in the drawer and pulls out a red sock. He is wearing blue shorts, so he replaces it. He then draws out a white sock. What is the probability that Irvin pulls out a red sock, replaces it, and then pulls out a white sock?
First let’s draw a tree diagram.
There are 18 socks in Irvin’s sock drawer. The probability of getting a red sock when he pulls out the first sock is:
Irvin puts the sock back in the drawer and pulls out the second sock. The probability of getting a white sock on the second draw is:
Therefore, the probability of getting a red sock and then a white sock when the first sock is replaced is:
One important part of these types of problems is that order is not important.
Let’s say Irvin picked out a white sock, replaced it, and then picked out a red sock. Calculate this probability.
In Example 1, what happens if the first sock is not replaced?
The probability that the first sock is red is:
The probability of picking a white sock on the second pick is now:
So now, the probability of selecting a red sock and then a white sock, without replacement, is:
The probability of picking a red sock on the second pick is now: