Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Introduction to Probability and Statistics Twelfth Edition Chapter 4 Probability and Probability Distributions Some graphic screen captures from Seeing Statistics ® Some images © 2001-(current year)

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc.Example Toss a fair coin twice. What is the probability of observing at least one head (Event A)? H H 1st Coin 2nd Coin E i P(E i ) H H T T T T H H T T HH HT TH TT 1/4 P(at least 1 head) = P(A) = P(HH) + P(HT) + P(TH) = 1/4 + 1/4 + 1/4 = 3/4 P(at least 1 head) = P(A) = P(HH) + P(HT) + P(TH) = 1/4 + 1/4 + 1/4 = 3/4 A={HH, HT, TH} S={HH, HT, TH, TT}

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc.Example A bowl contains three M&Ms ®, two reds, one blue. A child selects two M&Ms at random. What is the probability that exactly two reds (Event A)? 1st M&M 2nd M&M E i P(E i ) r1b r1r2 r2b r2r1 1/6 P(A) = P(r1r2) + P(r2r1) = 1/6 +1/6 = 1/3 P(A) = P(r1r2) + P(r2r1) = 1/6 +1/6 = 1/3 r1 b b b br1 br2 r2 r1b A={r1r2, r2r1}

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Counting Rules If the simple events in an experiment are equally likely, we can calculate We can use counting rules to find #A and #S.

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc.Counting How many ways from A to C? 3  2 = 6 3  2  2 = 12 How many ways from A to D?

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. The mn Rule If an experiment is performed in two stages, with m ways to accomplish the first stage and n ways to accomplish the second stage, then there are mn ways to accomplish the experiment. This rule is easily extended to k stages, with the number of ways equal to n 1 n 2 n 3 … n k Example: Example: Toss two coins. The total number of simple events is: 2  2 = 4

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Examples Example: Example: Toss three coins. The total number of simple events is: 2  2  2 = 8 Example: Example: Two M&Ms are drawn from a dish containing two red and two blue candies. The total number of simple events is: 6  6 = 36 Example: Example: Toss two dice. The total number of simple events is: m m 4  3 = 12

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Permutations The number of ways you can arrange n distinct objects, taking them r at a time is The order of the choice is important! Example: Example: How many 3-digit lock combinations can we make by using 3 different numbers among 1, 2, 3, 4 and 5?

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Example Example: Example: A lock consists of five parts and can be assembled in any order. A quality control engineer wants to test each order for efficiency of assembly. How many orders are there? The order of the choice is important!

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Example How many ways to select a student committee of 3 members: chair, vice chair, and secretary out of 8 students? The order of the choice is important! ---- Permutation

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Example How many ways to select a student committee of 3 members out of 8 students? (Don’t assign chair, vice chair and secretary). The order of the choice is NOT important!  Combination

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc.Combinations The number of distinct combinations of n distinct objects that can be formed, taking them r at a time is Example: Example: Three members of a 5-person committee must be chosen to form a subcommittee. How many different subcommittees could be formed? The order of the choice is not important!

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Example How many ways to select a student committee of 3 members out of 8 students? (Don’t assign chair, vice chair and secretary). The order of the choice is NOT important!  Combination

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Question A box contains six M&Ms ®, 4 reds and 3 blues. A child selects three M&Ms at random. What is the probability that exactly one is red (Event A) ? r1r4r3r2b2b1b3 Simple Events and sample space S: {r1r2r3, r1r2b1, r2b1b2…... } Simple events in event A: {r1b1b2, r1b2b3, r2b1b2……}

Copyright ©2006 Brooks/Cole A division of Thomson Learning, Inc. Solution Choose 3 MMs out of 7. (Total number of ways, i.e. size of sample space S) The order of the choice is not important! 4  3 = 12 ways to choose 1 red and 2 greens ( mn Rule) Event A: one red, two blues Choose one red Choose Two Blues