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

The Practice of Statistics, 5 th Edition2 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. 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 Edition3 Calculating Conditional Probabilities To find the conditional probability P(A | B), use the formula Calculating Conditional Probabilities

The Practice of Statistics, 5 th Edition4 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 = 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 Edition5 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 Edition6 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 Edition7 Example: Tree Diagrams Tennis great Roger Federer made 63% of his first serves in the 2011 season. When Federer made his first serve, he won 78% of the points. When Federer missed his first serve and had to serve again, he won only 57% of the points. 1.Draw a tree diagram to represent this situation. 2.What’s the probability that Federer makes the first serve and wins the point? 3.When he is serving, what’s the probability that he wins the point? 4.What is the probability he missed the first serve given he won the point?

The Practice of Statistics, 5 th Edition8 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. If two events are mutually exclusive, they cannot be 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).

The Practice of Statistics, 5 th Edition9 Checking for Independence Is there a relationship between gender and handedness? The two-way table displays data on the gender and dominant hand of each student. Gender Dominant HandMaleFemaleTotal Right Left Total Are the events “male” and “left-handed” independent? First determine P(left-handed|male), then determine P(left-handed). Are the probabilities the same or different? (Or you could have done P(male) and P(male|left-handed)

The Practice of Statistics, 5 th Edition10 Check your understanding For each chance process below, determine whether the events are independent. 1.Shuffle a standard deck of cards, and turn over the top card. Put it back in the deck, shuffle again, and turn over the top card. Define events A: first card is a heart, and B: second card is a heart. 2.Shuffle a standard deck of cards, and turn over the top two cards, one at a time. Define events A: first card is a heart, and B: second card is a heart.

The Practice of Statistics, 5 th Edition11 Check your understanding Determine whether the events are independent. The 28 students in Mr. Tabor’s AP Stat class completed a brief survey about right- or left-handed. The two-way table summarizes the data. Choose a student from the class at random. The events of interest are “female” and “right- handed”. Gender Dominant HandMaleFemaleTotal Left 13 4 Right Total

The Practice of Statistics, 5 th Edition12 Conditional Probability and Independence 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)

The Practice of Statistics, 5 th Edition13 Check your understanding 1.During World War II, the British found that the probability that a bomber is lost through enemy action on a mission over occupied Europe was Assuming that missions are independent, find the probability that a bomber returned safely from 20 missions. 2.Government data show that 8% of adults are full-time college students and that 30% of adults are age 55 or older. Because (0.08)(0.30) = 0.024, can we conclude that about 2.4% of adults are college students 55 or older? Why or why not?