5 3 Conditional Probability and Independence Objectives SWBAT

5. 3 Conditional Probability and Independence Objectives SWBAT: • 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.

In 2012, fans at Arizona Diamondbacks home games would win 3 free tacos from Taco Bell if the Diamondbacks scored 6 or more runs. In the 2012 season, the Diamondbacks won 41 of their 81 home games and gave away free tacos in 30 of their 81 home games. In 26 of the games, the Diamondbacks won and gave away free tacos. Let W=win and T=free tacos. Choose a Diamondbacks home game at random. a) Summarize these data in a two-way table. b) Find the probability that the Dbacks win or there are free tacos. c) Find the probability that the Dbacks win, given that there are free tacos. d) Find the probability that there are free tacos, given that the Dbacks win.

e) What is conditional probability? Is there a formula? Is it on the formula sheet? The probability that one event happens given that another event is already known to have happened is called a conditional probability. On the formula sheet: 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” Let’s go back to the previous example and look how we can apply the formula. We take the probability of both events occurring and divide it by the probability of the given event occurring.

c) Find the probability that the Dbacks win, given that there are free tacos. d) Find the probability that there are free tacos, given that the Dbacks win.

Shannon hits the snooze bar on her alarm clock on 60% of school days. If she doesn’t hit the snooze bar, there is a 0. 90 probability that she makes it to class on time. However, if she hits the snooze bar, there is only a 0. 70 probability that she makes it to class on time. a) Use a tree diagram to represent this situation. Probabilities 0. 60(0. 70)=0. 42 0. 60(0. 30)=0. 18 0. 40(0. 90)=0. 36 0. 40(0. 10)=0. 04 b) On a randomly chosen day, what is the probability that Shannon is late for class? P(L)= 0. 18 + 0. 04 = 0. 22 c) Suppose that Shannon is late for school. What is the probability that she hit the snooze bar that morning?

What is the general multiplication rule? Is it on the formula sheet? 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. Note: this is not on the formula sheet. These end probabilities were all examples of the general multiplication rule. For example, the 0. 42 comes from the probability of hitting snooze multiplied by the probability of being on time given Shannon hit snooze.

When is it better to use a tree diagram than a two-way table? • Using a tree diagram is better when the chance process involves a sequence of outcomes. • Two-way tables do not display sequences. Many employers require prospective employees to take a drug test. A positive result on this test indicates that the prospective employee uses illegal drugs. However, not all people who test positive actually use drugs. Suppose that 4% of prospective employees use drugs, the false positive rate is 5%, and the false negative rate is 10%. a) What percent of prospective employees will test positive? P(test positive)=. 036+. 048=. 084 b) What percent of prospective employees who test positive actually use illegal drugs?

How can you tell if two events are independent? 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). For example, if a coin is tossed twice in succession, the probability of obtaining a head on the second toss is not in any way influenced by the outcome of the first toss. Therefore, the two tosses are independent of each other. P (A) = P (A | B) = P (A | not B) Using the coin example, the probability of getting heads on the second toss = the probability of getting heads on the second toss given the first toss came up heads = the probability of getting heads on the second toss given the first toss came up tails

In the Diamondbacks example, are the events Taco and Win independent? To check whether two events are independent, we need to check whether knowing the team won the game changes the probability that the fans were given free tacos. In other words, were fans more likely to get free tacos if the team won (not independent events) or is the likelihood of fans getting free tacos the same whether or not the team wins (independent events)? Clearly Diamondbacks fans are more likely to get free tacos if the Diamondbacks win than if they lose, so the events taco and win are NOT independent.

What is the multiplication rule for independent events? Is it on the formula sheet? How is it related to the general multiplication rule? 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) This is not on the formula sheet.

What is the difference between mutually exclusive and independent? • Two events are mutually exclusive if they have no outcomes in common. • Example: The probability of selecting one LHS student and having that student be both a sophomore and a senior. • Being independent means that one event occurring will not change the likelihood of another event occurring. • Example: If a stadium is handing out free cats to all patrons entering, then free cats and win would be independent because the outcome of the game will not effect whether or not a patron gets a free cat. • Mutually exclusive events can never be independent, because if one event happens, the other is guaranteed not to happen.

The First Trimester Screen is a non-invasive test given during the first trimester of pregnancy to determine if there are specific chromosomal abnormalities in the fetus. According to a study published in the New England Journal of Medicine in November 2005, approximately 5% of normal pregnancies will receive a false positive result. Among 100 women with normal pregnancies, what is the probability that there will be at least one false positive? • It is reasonable to assume that the test results for different women are independent. • In order to find the probability of at least one false positive, it would probably be easiest to find the probability of no positive results at all. Then, the probability of at least one false positive would be the complement of this event: • P(at least one positive) = 1 – P(no positive results) • This works because “at least one false positive” would include anywhere between 1 -100 false positives. The only other possibility is no false positives, which is the complement event. • For women with normal pregnancies, the probability that a single test is negative is 1 – 0. 05 = 0. 95. • The probability that all 100 women will get negative results is (0. 95) = (0. 95) 100 = 0. 0059. • Thus, P(at least one positive) = 1 – 0. 0059 = 0. 9941. There is over a 99% probability that at least 1 of the 100 women with normal pregnancies will receive a false positive on the First Trimester Screen.

On a recent day, the Carucci Daily Cat forecast a 50% chance of rain in Lyndhurst and a 50% chance of rain in North Arlington. What is the probability it will rain in both locations? Would it be appropriate to perform the calculation (0. 5) = 0. 25, and state that the probability that it will rain in both cities tomorrow is 0. 25? No. It is not appropriate to multiply the two probabilities, because the events aren’t independent. If it is raining in one of these locations, there is a very high probability that it is raining in the other location. However, suppose that there was also a 50% chance of rain in Cupertino tomorrow. To find the probability that it will rain in Cupertino and in Lyndhurst, it would be appropriate to multiply the probabilities, because it is reasonable to believe that knowledge of rain in Lyndhurst won’t help us predict rain in Cupertino.

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