Tree Diagrams

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.

Using a Tree Diagram to Find Probability

Example 1

If you toss a coin 2 times, what is the probability of getting 2 heads? Use a tree diagram to find your answer.

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.

From the tree diagram, you can see that the probability of getting a head on the first flip is 1/2. Starting with heads, the probability of getting a second head will again be  1/2. But how do we calculate the probability of getting 2 heads? These are independent events, since the outcome of tossing the first coin in no way affects the outcome of tossing the second coin. Therefore, we can calculate the probability as follows:

P(A and B) = 1/2 x 1/2 = 1/4

Therefore, we can conclude that the probability of getting 2 heads when tossing a coin twice is , or 25%.

Example 2

Irvin is getting dressed for school. He knows his sock drawer contains 4 red socks, 6 white socks, and 8 brown socks, but they are all loose and unsorted. Because the room is dim, he cannot distinguish the colors as he reaches in.

Irvin pulls out one sock at random, sees that it is red, and decides it doesn't match his blue shorts. He replaces the red sock in the drawer and mixes them up. He then reaches in a second time and pulls out a white sock.

What is the probability of this specific sequence: pulling a red sock, replacing it, and then pulling a white sock?

There are 18 socks in the drawer. The probability of getting a red sock in the first draw is :

P(red) = 4/18 

Irvin puts the sock (replacement) and draws a second sock. The probability of getting a white sock on the second draw is:

P(white) = 6/18

Therefore, the probability of getting a red sock and then a white sock when the first sock is replaced is:

P(red and white) = 4/18 x 6/18 = 24/324 = 2/27

Note that since the first sock was replaced, each draw of a sock is independent of each other

Example 3 

In the previous example, what happens if the first sock is not replaced? Since the sock is not replaced, we say the second draw is dependent on the first draw. 

The probability of the first stock being red is unchanged:

P(red) = 4/18

But since the sock from the first draw was not replaced, there are only 17 socks left in the drawer. So the probability of picking a white sock on the second pick is now:

P(white|red) = 6/17

Now the probability of selecting a red sock and then a white sock, without replacement, is:

P(red and white) = 4/18 x 6/17 = 24/306 = 4/51

CK-12: Tree Diagrams

LibreTexts: Probability Using Tree Diagrams and Combinations

https://mathleaks.com/study/kb/concept/tree_diagram