Conditional Probability Brenda Meery Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org AUTHORS Brenda Meery CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform. Copyright 2012 CK-12 Foundation, www.ck12.org The names CK-12 and CK12 and associated logos and the terms FlexBook and FlexBook Platform (collectively CK-12 Marks ) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non- Commercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: June 22, 2012
www.ck12.org 1 CONCEPT 1 Conditional Probability Here you ll learn the definition of conditional probability and how to use conditional probability to solve for probabilities in finite sample spaces. You ve just done some data collection to determine the popularity of courses at your high school. Your calculations show that 75% of the students take Geometry and 15% of the students take both Chemistry and Geometry. How would you find the probability that a student who is taking Chemistry is also taking Geometry? Watch This MEDIA Click image to the left for more content. Khan Academy Probability (part 6) Guidance What if the probability of a second event is affected by the probability of the first event? This type of probability calculation is known as conditional probability. When working with events that are conditionally probable, you are working with 2 events, where the probability of the second event is conditional on the first event occurring. Say, for example, that you want to know the probability of drawing 2 kings from a deck of cards. As we have previously learned, here is how you would calculate this: P(first king) = 1 13 P(second king) = 3 51 P(2 kings) = 1 13 3 51 P(2 kings) = 3 663 P(2 kings) = 1 221 Now let s assume you are playing a game where you need to draw 2 kings to win. You draw the first card and get a king. What is the probability of getting a king on the second card? The probability of getting a king on the second card can be thought of as a conditional probability. The formula for calculating conditional probability is given as: P(B A) = P(A B) P(A) P(A B) = P(A) P(B A) Concept 1. Conditional Probability
2 www.ck12.org Another way to look at the conditional probability formula is as follows. Assuming the first event has occurred, the probability of the second event occurring is: P(second event first event) = P(first event and second event) P(first event) Let s work through a few problems using the formula for conditional probability. Example A You are playing a game of cards where the winner is determined when a player gets 2 cards of the same suit. You draw a card and get a club ( ). What is the probability that the second card will be a club? Step 1: List what you know. First event = drawing the first club Second event = drawing the second club P(first club) = 13 52 P(second club) = 12 51 P(club and club) = 13 52 12 51 P(club and club) = 156 2,652 P(club and club) = 1 17 Step 2: Calculate the probability of choosing a club as the second card when a club is chosen as the first card.
www.ck12.org 3 P(club and club) Probability of drawing the second club = P(first club) P(club club) = 1 17 13 52 P(club club) = 1 17 52 13 P(club club) = 52 221 P(club club) = 4 17 Step 3: Write your conclusion. Therefore, the probability of selecting a club as the second card when a club is chosen as the first card is 24%. Example B In the next round of the game, the first person to be dealt a black ace wins the game. You get your first card, and it is a queen. What is the probability of obtaining a black ace? Step 1: List what you know. First event = being dealt the queen Second event = being dealt the black ace P(queen) = 4 52 P(black ace) = 2 51 P(black ace and queen) = 4 52 2 51 P(black ace and queen) = 8 2,652 P(black ace and queen) = 2 663 Step 2: Calculate the probability of choosing black ace as a second card when a queen is chosen as a first card. Concept 1. Conditional Probability
4 www.ck12.org Step 3: Write your conclusion. P(black ace and queen) P(black ace queen) = P(queen) P(black ace queen) = 2 663 4 52 P(black ace queen) = 2 663 52 4 P(black ace queen) = 104 2,652 P(black ace queen) = 2 51 Therefore, the probability of selecting a black ace as the second card when a queen is chosen as the first card is 3.9%. Example C Sandra went out for her daily run. She goes on a path that has alternate routes to give her a variety of choices to make her run more enjoyable. The path has 3 turns where she can go left or right at each turn. The probability of turning right the first time is 1 2. Based on past runs, the probability of turning right the second time is 2 3. Draw a tree diagram to represent the path. What is the probability that she will turn left the second time after turning right the first time? Step 1: List what you know. P(right the first time) = 1 2 P(right the second time) = 2 3 P(left the second time) = 1 2 3 = 1 3 P(right the first time and left the second time) = 1 2 1 3 P(right the first time and left the second time) = 1 6
www.ck12.org 5 Step 2: Calculate the probability of choosing left as the second turn when right is chosen as the first turn. P(right the first time and left the second time) P(left the second time right the first time) = P(right the first time) P(left the second time right the first time) = P(left the second time right the first time) = 1 6 2 1 1 6 1 2 P(left the second time right the first time) = 2 6 P(left the second time right the first time) = 1 3 P(left the second time right the first time) = 0.333 P(left the second time right the first time) = 33% Step 3: Write your conclusion. Therefore, the probability of choosing left as the second turn when right was chosen as the first turn is 33%. Vocabulary The probability of a particular dependent event, given the outcome of the event on which it depends, is known as conditional probability. Guided Practice At Bluenose High School, 90% of the students take Physics and 35% of the students take both Physics and Statistics. What is the probability that a student from Bluenose High School who is taking Physics is also taking Statistics? Answer: Step 1: List what you know. P(physics) = 0.90 P(physics and statistics) = 0.35 Step 2: Calculate the probability of choosing Statistics as a second course when Physics is chosen as a first course. Concept 1. Conditional Probability
6 www.ck12.org P(physics and statistics) P(statistics physics) = P(physics) P(statistics physics) = 0.35 0.90 P(statistics physics) = 0.388 P(statistics physics) = 39% Step 3: Write your conclusion. Therefore, the probability that a student from Bluenose High School who is taking Physics is also taking Statistics is 39%. Interactive Practice Practice 1. 2 fair dice are rolled. What is the probability that the sum is even given that the first die that is rolled is a 2? 2. 2 fair dice are rolled. What is the probability that the sum is even given that the first die rolled is a 5? 3. 2 fair dice are rolled. What is the probability that the sum is odd given that the first die rolled is a 5? 4. Steve and Scott are playing a game of cards with a standard deck of playing cards. Steve deals Scott a black king. What is the probability that Scott s second card will be a red card? 5. Sandra and Karen are playing a game of cards with a standard deck of playing cards. Sandra deals Karen a red seven. What is the probability that Karen s second card will be a black card? 6. Donna discusses with her parents the idea that she should get an allowance. She says that in her class, 55% of her classmates receive an allowance for doing chores, and 25% get an allowance for doing chores and are good to their parents. Her mom asks Donna what the probability is that a classmate will be good to his or her parents given that he or she receives an allowance for doing chores. What should Donna s answer be? 7. At a local high school, the probability that a student speaks English and French is 15%. The probability that a student speaks French is 45%. What is the probability that a student speaks English, given that the student speaks French? 8. At a local high school, the probability that a student takes statistics and art is 10%. The probability that a student takes art is 65%. What is the probability that a student takes statistics, given that the student takes art?
www.ck12.org 7 9. The test for a disease is accurate 80% of the time, and 2.5% of the population has the disease. What is the probability that you have the disease, given that you tested positive? 10. For question 9, what is the probability that you don t have the disease, given that you tested negative? Concept 1. Conditional Probability
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