The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers CHAPTER 5 Probability: What Are the Chances? 5.3 Conditional Probability and Independence

Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition2 CALCULATE and INTERPRET conditional probabilities. USE the general multiplication rule to CALCULATE probabilities. USE tree diagrams to MODEL a chance process and CALCULATE probabilities involving two or more events. DETERMINE if two events are independent. When appropriate, USE the multiplication rule for independent events to COMPUTE probabilities. Conditional Probability and Independence

The Practice of Statistics, 5 th Edition3 What is Conditional Probability? The probability we assign to an event can change if we know that some other event has occurred. This idea is the key to many applications of probability. When we are trying to find the probability that one event will happen under the condition that some other event is already known to have occurred, we are trying to determine a conditional probability. The probability that one event happens given that another event is already known to have happened is called a conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B | A). The probability that one event happens given that another event is already known to have happened is called a conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B | A). Read | as “given that” or “under the condition that”

The Practice of Statistics, 5 th Edition4 Calculating Conditional Probabilities To find the conditional probability P(A | B), use the formula The conditional probability P(B | A) is given by Calculating Conditional Probabilities

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6 Who Reads the Paper? What is the probability that a randomly selected resident who reads USA Today also reads the New York Times? There is a 12.5% chance that a randomly selected resident who reads USA Today also reads the New York Times.

The Practice of Statistics, 5 th Edition7 CYU What is the probability that a randomly selected household with a landline also has a cell phone? There is a 0.85 probability that a landline user also has a cell phone.

The Practice of Statistics, 5 th Edition8 Calculating Conditional Probabilities Find P(L) Find P(E | L) Find P(L | E) Total Total P(L) = 3656 / = P(E | L) = 800 / 3656 = P(L| E) = 800 / 1600 = Consider the two-way table on page 321. Define events E: the grade comes from an EPS course, and L: the grade is lower than a B.

The Practice of Statistics, 5 th Edition9 The General Multiplication Rule The probability that events A and B both occur can be found using the general multiplication rule P(A ∩ B) = P(A) P(B | A) where P(B | A) is the conditional probability that event B occurs given that event A has already occurred. General Multiplication Rule In words, this rule says that for both of two events to occur, first one must occur, and then given that the first event has occurred, the second must occur.

The Practice of Statistics, 5 th Edition10 Teens With Online Profiles The Pew Internet and American Life Project find that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. Find the probability that a randomly selected teen uses the Internet and has posted a profile. Show your work. There is about a 51% chance that a randomly selected teen uses the Internet and has posted a profile on a social-networking site.

The Practice of Statistics, 5 th Edition11 Tree Diagrams The general multiplication rule is especially useful when a chance process involves a sequence of outcomes. In such cases, we can use a tree diagram to display the sample space. Consider flipping a coin twice. What is the probability of getting two heads? Sample Space: HH HT TH TT So, P(two heads) = P(HH) = 1/4

The Practice of Statistics, 5 th Edition12 Example: Tree Diagrams 51.15% of teens are online and have posted a profile. The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. What percent of teens are online and have posted a profile?

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The Practice of Statistics, 5 th Edition14 What is the probability that Federer makes the first serve and wins the point? When Federer is serving, what is the probability that he wins the point?

The Practice of Statistics, 5 th Edition15 (a) Draw a tree diagram to represent this situation. (b) Find the probability that this person has visited a video-sharing site. Show your work. (c) Given that this person has visited a video-sharing site, find the probability that he or she is aged 18 to 29. Show your work.

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The Practice of Statistics, 5 th Edition19 CYU on p.326

The Practice of Statistics, 5 th Edition20 Conditional Probability and Independence When knowledge that one event has happened does not change the likelihood that another event will happen, we say that the two events are independent. Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B). Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B). When events A and B are independent, we can simplify the general multiplication rule since P(B| A) = P(B). Multiplication rule for independent events If A and B are independent events, then the probability that A and B both occur is P(A ∩ B) = P(A) P(B) Multiplication rule for independent events If A and B are independent events, then the probability that A and B both occur is P(A ∩ B) = P(A) P(B)

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The Practice of Statistics, 5 th Edition22 Testing for Independence The events of interest are A: is male and B: has pierced ears. For the two events to be independent, or Since the conditional and unconditional probabilities are not the same the two events are not independent.

The Practice of Statistics, 5 th Edition23 CYU on p.328

The Practice of Statistics, 5 th Edition24 Multiplication Rule for Independent Events Following the Space Shuttle Challenger disaster, it was determined that the failure of O-ring joints in the shuttle’s booster rockets was to blame. Under cold conditions, it was estimated that the probability that an individual O-ring joint would function properly was Assuming O-ring joints succeed or fail independently, what is the probability all six would function properly? P( joint 1 OK and joint 2 OK and joint 3 OK and joint 4 OK and joint 5 OK and joint 6 OK) By the multiplication rule for independent events, this probability is: P(joint 1 OK) · P(joint 2 OK) · P (joint 3 OK) … · P (joint 6 OK) = (0.977)(0.977)(0.977)(0.977)(0.977)(0.977) = 0.87 There’s an 87% chance that the shuttle would launch safely under similar conditions (and a 13% chance that it wouldn’t).

The Practice of Statistics, 5 th Edition25 Use the fact that “At least 1” and “none” are opposites in order to help set up this problem

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The Practice of Statistics, 5 th Edition28 CYU on p.331

Section Summary In this section, we learned how to… The Practice of Statistics, 5 th Edition29 CALCULATE and INTERPRET conditional probabilities. USE the general multiplication rule to CALCULATE probabilities. USE tree diagrams to MODEL a chance process and CALCULATE probabilities involving two or more events. DETERMINE if two events are independent. When appropriate, USE the multiplication rule for independent events to COMPUTE probabilities. Conditional Probabilities and Independence