Statistics Independent Study Page 1 of 3 Study Sheet Tree Diagrams and Probabilities About tree diagrams In a tree diagram, the simple events are on the ends of the branches, and the ends of all the branches make up your sample space. Branching occurs with every condition. Tree diagrams follow the addition and multiplication rules of probability. To calculate the probability of a simple event you multiply the probabilities of each segment along the path to that simple event. To calculate the probability of an event, add together the probabilities of all the simple events within that event. Questions 1. Using the below table, make a tree diagram representing the probabilities in this

table. Start by making the first set of branches on the left the political party. (Hint: Remember that the ends of the branches, the simple events, must sum to 1. Each set of branches should also sum to 1. Finally, only some of the numbers in the tree diagram are in the table, a few you have to calculate by hand.)

Voter support for political term limits is strong in many parts of the United States. A poll conducted by the Field Institute in California showed support for term limits by a 21 margin. the results of this poll of n = 347 registered voters are given in the table:

For (F) Against (A) No Opinion (N) Total Republican (R) .28 .10 .02 .40 Democrat (D) .31 .16 .03 .50 Other (O) .06 .04 .00 .10 Total .65 .30 .05 1.00 Data Source: Dion Nissenbaum, "Support Grows for Term Limits," The Press-Enterprise (Riverside, CA), 30 May 1997, p. A-1. 2. Using the tree diagram you just constructed, find the following probabilities:

A. The probability of being Republican B. The probability of being For C. The probability of being a Democrat and Against D. The probability of being a Democrat or Against E. The probability of being Against, given an individual is a Democrat F. The probability of having No opinion, given an individual is a Republican

3. Make a tree diagram as in Question 1, but this time start with poll results as the first

set of branches. 4. Using the tree diagram you constructed in Question 3, find the following

probabilities: A. The probability of having No opinion B. The probability of being Other C. The probability of being For and a Republican D. The probability of being For or a Republican E. The probability of being a Republican, given that one is Against

Statistics Independent Study Page 2 of 3 Study Sheet Tree Diagrams and Probabilities

F. The probability of being a Democrat, given that one is For 5. Look at the two tree diagrams you've constructed. What probabilities remain the

same regardless of which tree diagram you use? 6. Look at your answers to 2.A. and 4.E. Why are they different? Under what

circumstances would they have the same probability, and what would that probability be?

7. Fill in the missing values on this tree diagram.

8. Convert the tree diagram you just completed in Question 7 to a table. Put the colors

in the columns and the shapes in the rows. Don't forget to include the marginal probabilities in your table.

9. In this table, are color and shape independent? How can you tell? 10. If the marginal probabilities you got for the table in Question 8 remained the same,

and color and shape were independent, what would the table look like? (Show all probabilities to the nearest hundredth.)

Statistics Independent Study Page 3 of 3 Study Sheet Tree Diagrams and Probabilities Acknowledgements Question 1: This question uses the data table from question 4.63 from page 151 of Introduction to Probability and Statistics, Tenth Edition, by W. Mendenhall, R. Beaver, and B. Beaver. Copyright 1999 by Brooks Cole, division of Thompson Learning Incorporated. Further reproduction is prohibited without permission of the publisher. __________________ Copyright 2000 Apex Learning Inc. All rights reserved. This material is intended for the exclusive use of registered users only. No portion of these materials may be reproduced or redistributed in any form without the express written permission of Apex Learning Inc.