Question #4:
4) There is 1 red ball, 1 blue ball, and 1 yellow ball in a hat. Two balls are selected. Construct a tree diagram and list the sample space if the selection is done
a) with replacement.
b) without replacement.

Solution: 

First, let’s look at the information that is given to us by the problem.

• 1 Red Ball
• 1 Blue Ball
• 1 Yellow Ball
• 2 Balls are selected (with/without replacement)

In order to solve this problem, we first must recognize that it is a two part question. We must construct a tree diagram and list the sample space, but in two different ways. The first is with replacement, meaning that after we pull a ball out of the hat, we replace it within the hat. This just means that it remains available to be pulled from the hat again. The second way is without replacement, meaning that after the ball it pulled from the hat, it is not returned and it is no longer possible to select that ball.

This is a problem that is easier to understand with a visual aid. I have attached a photo of the tree diagram and sample size for with and without replacement. In order to make things easier, I color coded the tree diagram – but don’t get confused, I used green instead of yellow as it would have been more difficult to read.

Step 1: Let’s create the tree diagram and sample size for the trial with replacement first.

In order to do so, we first start with the root of the tree – a single point that separates into our first number of trials. The first trial accounts for each ball that can be grabbed the first time. One red, one blue, or one yellow. However, if you are replacing that ball, it can be grabbed again during the second selection. Therefore, each of the existing three colors should branch off into three more options. The three options for each of the existing three should also be one red, one blue, and one yellow. For example, because the ball is replaced, if you grab a red ball the first time, you can grab a red ball again the second time, etc. There should be nine options total in the second trial, three for each of the three from the first trial.

The sample size can be listed from the tree diagram. For example, just read through each branch of the diagram and account for every possible selection.

Step 2: In order to create the tree diagram for each trial without replacement, after the first trial, omit the same color ball for the second trial. This means that because the balls are not  replaced, you can only grab the first color once and there is no chance of grabbing it again. To explain further, once again you will have three balls for your first trial – one red, one blue, and one yellow. However, because you are not replacing the ball, each second trial will only branch off into two other options – the colors that were NOT grabbed on the first trial. So, instead of nine options for the second trial, three are no longer an option, meaning that there should be six. If this doesn’t make sense, take a look at the picture I attached – as I said, it is much easier to understand this sort of problem with help of a visual aid.

If there are any questions, feel free to ask and I will post a reply in attempt to clarify.