# Probability Notes

## Using tree diagrams: an example

Abstract: An example showing how a tree diagram may be used to find the probabilities of various events that may occur when colored beads are drawn from a jar.

### Problem

A jar contains 2 red, 2 green and 1 blue beads. Two beads are drawn without replacement. Use a tree diagram to illustrate the outcomes and figure out the probability of drawing at least one red bead.

### Solution

Here is a "tree diagram" for this problem. The fractions in parentheses give the probabilities a bead of the indicated color being drawn at each stage. For example, the figure (2/5) after "Red" in the "First Draw" column comes from the fact that at this stage there are 2 red beads out of 5 beads all together in the jar. The figure (1/4) in the top box in the "Second Draw" column comes from the fact that now, after one red has been removed, there is only 1 red of 4 beads.

First Draw
 Second Draw Outcome Probability
Red (2/5)
 Red (1/4) RR (2/5)(1/4)=1/10 Green (2/4) RG (2/5)(2/4)=1/5 Blue (1/4) RB (2/5)(1/4)=1/10
Green (2/5)
 Red (2/4) GR (2/5)(2/4)=1/5 Green (1/4) GG (2/5)(1/4)=1/10 Blue (1/4) GB (2/5)(1/4)=1/10
Blue (1/5)
 Red (2/4) BR (1/5)(2/4)=1/10 Green (2/4) BG (1/5)(2/4)=1/10

 The event "at least one bead is red" is indicated in pink. The sum of the probabilities of the outcomes in this event is 7/10.

### Test yourself

1. What is the probability of drawing two beads of different colors?
2. List some other events and find their probability.
3. Modify the problem by considering other combinations of beads.

### Going deep

This problem has been solved using a tree model. It can also be solved with a "sample space" model in which the outcomes are equally likely. This requires that we take a different view of the outcomes by giving each bead a separate identity. The sample space then contains outcomes such as: R2G2, meaning that red ball number 2 and green ball number 2 have been chosen. A list of all outcomes would begin: R1R2, R1G1, R1G2, R1B, R2R1,R2G1, etc. Problem: Work out the details of a solution using this model, and show that the same probabilities are obtained for the events you considered.