Section 5.3 Conditional Probability and Independence Learning Objectives After this section, you should be able to… DEFINE conditional probability COMPUTE conditional probabilities DESCRIBE chance behavior with a tree diagram DEFINE independent events DETERMINE whether two events are independent APPLY the general multiplication rule to solve probability questions

Conditional Probability and Independence 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. ****Knowing something will usually increase the probability Definition: 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).

Conditional Probability and Independence Example: Use the table to find the probabilities. An SRS of students at a very large high school were asked whether they felt their teachers cared about them What’s the probability we randomly select someone who said “Yes”? What’s the probability we select someone at random who said yes, if they were agirl? P(Yes 10 th grader)= P(No Senior)= P(Senior No)= P(10 th or No)= 9 th 10 th 11 th 12 th Total Yes No Total

Conditional Probability and Independence Don’t ever assume independence- Show the work!!!!! Yes, rolling a dice and flipping a coin are independent, but show it!!!! Example: A certain type of license plate has four spaces that can be numbers or letters, and repeatsare allowed. Are the events having all numbers and having all letters independent? Definition: Two events A and B are independent if the occurrence of one event has no effect on the chance 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).

Conditional Probability and Independence General Multiplication Rule The idea of multiplying along the branches in a tree diagram leads to a general method for finding theprobability P ( A ∩ B ) that two events happen together. 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

Conditional Probability and Independence Tree Diagrams—Can be very useful!!!! Draw one!!! A prominent university decides admissions based on the following criteria: Students with SATs under 1400 are rejected Those who are still considered will have their GPA examined. Those with GPAs over3.7 (unweighted) are accepted. Those under 3.2 are rejected. Those with GPAs between 3.2 and 3.7 will have their essays read by two admissionsofficers. If both like it, then they are accepted. Over the years the following statistics have been compiled: On average, 52% of applicants have SATs over Of the students who have over 1400 SAT scores: 61% have GPAS 3.7 or above 31% have GPAs between 3.2 and 3.7 There is a fifty-fifty shot an admissions officer will like the essay

Using the tree diagram- Answer the following questions: 1.What’s the probability a student is accepted? 2.What is the probability that a student who was accepted had a GPA of 3.7 or above? 3.What is the probability that a student who was accepted got in because of their essay? 4.What is the probability that if a student was rejected, it was because of the essays? 5.What is the probability a student got accepted if their SAT was 1400 or better?

Conditional Probability and Independence Example: Teens with Online Profiles 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?

Conditional Probability and Independence Independence: A Special Multiplication Rule When events A and B are independent, we can simplify the general multiplication rule since P ( B | A ) = P ( B ). Definition: 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) 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? Example:

Conditional Probability and Independence Calculating Conditional Probabilities If we rearrange the terms in the general multiplication rule, we can get a formula for the conditional probability P ( B | A ). P(A ∩ B) = P(A) P(B | A) General Multiplication Rule To find the conditional probability P(B | A), use the formula Conditional Probability Formula P(A ∩ B)P(B | A)P(A)P(A) = This is the way it is given on the AP Exam…..

Conditional Probability and Independence Example: 1. On September11th, 2002, the first anniversary of the terrorist attacks on the WorldTrade Center, the New York State Lottery’s daily number came up Is itcoincidence or something more? a) What is the probability that the three winning numbers match a date on any given day? b) What is the probability the whole year passes without this happening? c) What is the probability that the date and winning lottery number match at least once? d) If every one of the fifty states has a three-digit lottery, what is the probability at least one ofthem will come up on September 11 th ?

Conditional Probability and Independence Example: 2. A pharmacy has just come out with a new pregnancy test that registers blue (positiveresult) in 95% of users who are pregnant. However, the new test also registers bluein 5% of users who are not pregnant (false-positive). Suppose that, in reality, only4% of women using this test are pregnant. a) What is the probability that a randomly selected woman who uses this test gets a blue result? b) What is the probability that the woman is pregnant if the test registers blue? c) What is the probability that the woman is not pregnant, given a negative result?

Conditional Probability and Independence Example: 3. This is data from a recent year that gives auto crash fatality. Choosing at random! a) What is the probability that a randomly selected crash involved a single vehicle? b) What is the probability that a crash a single vehicle and alcohol? c) What is the probability that a crash involved a single vehicle or was alcohol related? d) What is the probability that a randomly chosen auto fatality was alcohol related, if itinvolved a single vehicle? e) Are the events that a fatal crash was alcohol related and a fatal crash involved a singlevehicle independent? Explain. Single VehicleMultiple VehicleTotal Alcohol Related10,7414,88715,628 No Alcohol11,34511,33622,681 Total22,08616,22338,309

Conditional Probability and Independence Example: 4. The Neptune Diner in Lancaster, PA employs three dishwashers. Alberto washes40% of the dishes and breaks about 1% of those he handles. Betty Sue and Chuckeach wash 30% of the dishes, and Betty Sue breaks only 2% of her dishes whilegood old Chuck breaks about 6% of his dishes. a) Why is Chuck still employed there? b) When you go to dinner there, you hear a dish break in the back. After clapping, what is theprobability that Chuck is washing dishes? c) What is the probability that it is not Chuck in the back?