Chapter 3: Probability Topics
3.5 Tree and Venn Diagrams
Learning Objectives
By the end of this section, you should be able to:
- Use tree diagrams to find probabilities for experiments with and without replacement
- Use Venn diagrams to visualize relationships between events
Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams and Venn diagrams are two tools that can be used to visualize and solve conditional probabilities.
Tree Diagrams
A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of “branches” that are labeled with either frequencies or probabilities. Tree diagrams can make some probability problems easier to visualize and solve. The following example illustrates how to use a tree diagram.
Example
In an urn, there are 11 balls. Draw two balls, one at a time, with replacement, which means that you put the first ball back in the urn before you select the second ball. The tree diagram using frequencies that show all the possible outcomes follows.
The first set of branches represents the first draw, where there are 8 ways to first draw B and three ways to first draw R. The second set of branches represents the second draw, with the same potential outcomes for each of the original branches due to the replacement of the first draw. Each of the outcomes is distinct. In fact, we can label each red ball as R1, R2, and R3 and each blue ball as B1, B2, B3, B4, B5, B6, B7, and B8. Then the nine RR outcomes can be written as:
{ R1R1, R1R2, R1R3, R2R1, R2R2, R2R3, R3R1, R3R2, R3R3}. Counting these shows 9 ways we can draw two red balls. We can also get this total from our tree diagram by multiplying down the branches: 3 times 3 = 9 RR outcomes.
Then the experiment to draw two balls, one at a time, with replacement has (11)(11) = 121 outcomes, the size of the sample space.
We can now use our tree diagram to compute some probabilities.
- Calculate [latex]P(RR)[/latex].
- Calculate [latex]P(RB \text{ OR } BR).[/latex]
- Calculate [latex]P(RB).[/latex]
- Calculate [latex]P(R\text{ 2nd } | B \text{ 1st} ).[/latex]
- Calculate [latex]P(BB).[/latex]
- Calculate [latex]P(B\text{ 2nd } | R \text{ 1st} ).[/latex]
Solution
b. P(RR) = [latex]\left(\frac{3}{11}\right)\left(\frac{3}{11}\right)[/latex] = [latex]\frac{9}{121}[/latex]
c. Using the tree diagram, calculate P(RB OR BR).
Solution
c. P(RB OR BR) = [latex]\left(\frac{3}{11}\right)\left(\frac{8}{11}\right)[/latex] + [latex]\left(\frac{8}{11}\right)\left(\frac{3}{11}\right)[/latex] = [latex]\frac{48}{121}[/latex]
d. Using the tree diagram, calculate P(R on 1st draw AND B on 2nd draw).
Solution
d. P(R on 1st draw AND B on 2nd draw) = P(RB) = [latex]\left(\frac{3}{11}\right)\left(\frac{8}{11}\right)[/latex] = [latex]\frac{24}{121}[/latex]
e. Using the tree diagram, calculate P(R on 2nd draw GIVEN B on 1st draw).
Solution
e. P(R on 2nd draw GIVEN B on 1st draw) = P(R on 2nd|B on 1st) = [latex]\frac{24}