## CONSTRUCTING A PROBABILITY DISTRIBUTION BY LISTING OUTCOMES

In a random experiment, the sample space is the collection of all possible outcomes. In some situations, it is reasonable to assume that all of the possible outcomes of the experiment are equally likely. In this case, it is straightforward to compute the probability distribution for some variable of interest.

Let us illustrate this construction process for a simple example. Suppose a room contains two men and three women. You wish to select two people from this class to serve on a committee. How many women will be on this committee? We don't know -- the number of women in the committee could be 0, 1 or 2. We are interested in obtaining the probability of each of the three possibilities.

First, we will represent the people in the room using the symbols

 W1 W2 W3 M1 M2

In the above, W represents a women and M a man and we distinguish between the people of the same sex.

Our experiment is selecting two people to serve on the committee. Using our symbols for the people, there are the following 10 possible committees. Note that we don't care what order the two people are selected; we are only interested in the group of people in the committee. If we select the committee at random, then each possible group of two people has the same chance of being selected. Since there are 10 groups, we assign to each possible committee the probability 1/10.

COMMITTEE PROBABILITY
W1, W2 1/10
W1, W3 1/10
W2, W3 1/10
M1, W1 1/10
M1, W2 1/10
M1, W3 1/10
M2, W1 1/10
M2, W2 1/10
M2, W3 1/10
M1, M2 1/10

Remember our interest was in the number of women on the committee. For each committee listed above, we can list the number of women selected. For example,in the committee {W1, W2}, 2 women were selected, for the committee {M2, W3}, 1 woman was selected, and so on. We put the number of women next to the group name in the table.

COMMITTEE # OF WOMEN PROBABILITY
W1, W2 2 1/10
W1, W3 2 1/10
W2, W3 2 1/10
M1, W1 1 1/10
M1, W2 1 1/10
M1, W3 1 1/10
M2, W1 1 1/10
M2, W2 1 1/10
M2, W3 1 1/10
M1, M2 0 1/10

Now we are ready to construct our probabilty table for "number of women". In the table below, we list all possible numbers of women we could pick (0, 1 or 2). Then we assign probabilities to the three outcomes by using the above table.

What is the probabilty that 0 women are selected. Looking at the table, we see that 0 women means that the committee selected was {M1, M2} which has probability 1/10. So the probability of 0 women is 1/10.

What is the probabilty that exactly 1 women is selected? Looking at the table, we see that we select exactly 1 women when the committes {M1, W1}, {M1, W2}, {M1, W3}, {M2, W1}, {M2, W2}, {M2, W3} are chosen. By adding the probabilities of the six outcomes, we see that the probability of 1 women is 6/10. It should be easy for you to find the probability that two women are selected. Putting this all together, we arrive at the following probability distribution for number of women.

# OF WOMEN PROBABILITY
0 1/10
1 6/10
2 3/10

Page Author: Jim Albert (© 1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/prob_list.html