Math 1100, Section 3 

Probability Notes 

Number 2 
10/9/2002 
by J. Madden
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.
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.
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 


Red (2/5) 


Green (2/5) 


Blue (1/5) 

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. 
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.