Probability and Combinatorics Chapter 8 Probability 8.3 Probability and Combinatorics MATHPOWERTM 12, WESTERN EDITION 8.3.1

Applications of Combinatorics to Probability Two cards are picked at random from a deck of playing cards. What is the probability that they are both kings? Method 1: Using the fundamental counting principle P(2 kings) = Method 2: Using combinations Number of ways of choosing 2 kings from 4 kings is 4C2. Number of ways of choosing 2 cards from 52 cards is 52C2. P(2 kings) = P(2 kings) 8.3.2

Applications of Combinatorics to Probability [cont’d] Method 3: Using permutations Number of ways of arranging 2 kings from 4 kings is 4P2. Number of ways of arranging 2 cards from 52 cards is 52P2. P(2 kings) = 8.3.3

Using Combinatorics to Calculate Probability 1. A committee of 4 is to be selected from a group of 7 women and 5 men. What is the probability that there will be at least 1 woman on the committee? Consider the compliment for these types of problems: P(at least 1 woman) = 1 - P(0 women) The number of ways of selecting 0 women and 4 men from the group is 7C0 x 5C4. The number of ways of selecting a committee of 4 from the group of 12 is 12C4. P(at least 1 woman) = 1 - P(0 women) = 1 - 0.010 = 0.99 The probability of at least 1 woman on the committee is 0.99. 8.3.4

Using Combinatorics to Calculate Probability 2. The 7 letters of the word KITCHEN are arranged randomly. What is the probability of an arrangement containing the letters KIT in a group but not necessarily in that order? The number of ways of arranging the letters of KITCHEN containing the letter group KIT is 5! x 3!. The number of ways of arranging the letters of KITCHEN in any order is 7! . P(event) = = 0.143 The probability of the arrangement containing the sequence KIT is 0.143. 8.3.5

Assignment Suggested Questions: Pages 385 and 386 1-5, 9-16, 11 ad, 12 ade, 13, 15 8.3.6