Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Probability 5

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section Conditional Probability and the General Multiplication Rule 5.4

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Compute conditional probabilities 2.Compute probabilities using the General Multiplication Rule 5-3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Compute Conditional Probabilities 5-4

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Conditional Probability The notation P(F|E) is read “the probability of event F given event E”. It is the probability that the event F occurs given that event E has occurred. 5-5

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE An Introduction to Conditional Probability Suppose that a single six-sided die is rolled. What is the probability that the die comes up 4? Now suppose that the die is rolled a second time, but we are told the outcome will be an even number. What is the probability that the die comes up 4? First roll: S = {1, 2, 3, 4, 5, 6} Second roll: S = {2, 4, 6} 5-6

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Conditional Probability Rule If E and F are any two events, then The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the number of outcomes in E. Conditional Probability Rule If E and F are any two events, then The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the number of outcomes in E. 5-7

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Conditional Probabilities on Belief about God and Region of the Country A survey was conducted by the Gallup Organization conducted May 8 – 11, 2008 in which 1,017 adult Americans were asked, “Which of the following statements comes closest to your belief about God – you believe in God, you don’t believe in God, but you do believe in a universal spirit or higher power, or you don’t believe in either?” The results of the survey, by region of the country, are given in the table on the next slide. 5-8

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Conditional Probabilities on Belief about God and Region of the Country Believe in God Believe in universal spirit Don’t believe in either East Midwest South West (a)What is the probability that a randomly selected adult American who lives in the East believes in God? (b)What is the probability that a randomly selected adult American who believes in God lives in the East?

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Conditional Probabilities on Belief about God and Region of the Country Believe in God Believe in universal spirit Don’t believe in either East Midwest South West

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Conditional Probabilities on Belief about God and Region of the Country Believe in God Believe in universal spirit Don’t believe in either East Midwest South West

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old. Also in 2005, 16.6% of all murder victims were 20 – 24 year old males. What is the probability that a randomly selected murder victim in 2005 was male given that the victim is 20 – 24 years old? EXAMPLE Murder Victims 5-12

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Compute Probabilities Using the General Multiplication Rule 5-13

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. General Multiplication Rule The probability that two events E and F both occur is In words, the probability of E and F is the probability of event E occurring times the probability of event F occurring, given the occurrence of event E. General Multiplication Rule The probability that two events E and F both occur is In words, the probability of E and F is the probability of event E occurring times the probability of event F occurring, given the occurrence of event E. 5-14

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old. Also in 2005, 86.9% of murder victims were male given that the victim was 20 – 24 years old. What is the probability that a randomly selected murder victim in 2005 was a 20 – 24 year old male? EXAMPLE Murder Victims 5-15