Lesson Objectives At the end of the lesson, students can: Identify and calculate conditional probabilities. Determine if two events are independent. Use tree diagrams to determine conditional probabilities. State and use the General Multiplication Rule.

Remember Key Words in Probability. A. “NOT” – subtract from one B Remember Key Words in Probability!!! A. “NOT” – subtract from one B. “OR” – add C. “AND” – multiply

Conditional Probability The probability of an event can change if we know that some other event has occurred. The probability that one event happens given that another event is already known to have happened is called conditional probability. The probability that event B happens given that event A has already happened is shown as P(B │ A).

Conditional Probability Pierced Ears? Gender Yes No Total Male 19 71 90 Female 84 4 88 103 75 178 Define event A: is male and B: has pierced ears for a randomly selected student. What is the probability that the student is male? P(A) = 90/178 What is the probability that the student has pierced ears? P (B) = 103/178 What is the probability that the student is male and has pierced ears? P(A∩B) = 19/178 What is the probability that the student is male or has pierced ears? P(AUB) = 90/178 + 103/178 – 19/178 = 174/178

Conditional Probability Pierced Ears? Gender Yes No Total Male 19 71 90 Female 84 4 88 103 75 178 If we know that a randomly selected student has pierced ears, what is the probability that the student is male?   If we know that the randomly selected student is male, what is the probability that the student has pierced ears? These questions sound the same, but are asking VERY different things! P(A│B) = 19/103 P(B│A) = 19/90

Conditional Probability Students at the University of New Harmony received 10,000 course grades last semester. The two-way table below breaks down these grades by which school of the university taught the course. The schools are Liberal Arts, Engineering and Physical Sciences, and Health and Human Services.   Grade Level School A B Below B Liberal Arts 2,142 1,890 2,268 Engineering and Physical Sci 368 432 800 Health and Human Services 882 630 588

Conditional Probability   Grade Level School A B Below B Liberal Arts 2,142 1,890 2,268 Engineering and Physical Sci 368 432 800 Health and Human Services 882 630 588 What is the probability that a grade is not A? What is the probability that a grade is B or better? What is the probability that a grade is an A, given that it comes from an HHS course? What is the probability that a grade is not B, given that it comes from an EPS course? What is the probability that a grade is A or below B, given that it comes from an LA course?

Conditional Probability   Grade Level School A B Below B Liberal Arts 2,142 1,890 2,268 Engineering and Physical Sci 368 432 800 Health and Human Services 882 630 588 Consider two events E: the grade comes from an EPS course, and L: the grade is lower than a B. Find P(L) Find P(E | L) Find P(L | E) Which of these conditional probabilities tells you whether this college’s EPS students tend to earn lower grades than students in liberal arts and social sciences? Explain. = 0.3656, which means 36.56% of grades are lower than a B = 800/3656 = 0.219 = 800/1600 = 0.5 P(L│E) – this is GIVEN the EPS students, the probability of having below a B. We could compare this to the other schools’ probability of having below a B.

Independent Events Two events A and B are independent if the occurrence of one event has no effect on the chance that the other even will happen. If two events are independent, then: P(A|B) = P(A) Given that event B happens, the probability of event A doesn’t change! P(B|A) = P(B) Given that event A happens, the probability of event B doesn’t change!

Are the Events Independent? 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: the first card is a ♥, and B: the second card is a ♥. Shuffle a standard deck of cards, and turn over the top two cards, one at a time. Define events A: the first card is a ♥, and B: the second card is a ♥. Independent. Since we are putting the first card back and then reshuffling the cards before drawing the 2nd card, knowing that the first card was will not tell us what the 2nd card will be. Not independent. Once we know the suit of the first card, then the probability of getting a heart on the second card will change, depending on what the first card was.

Are the Events Independent? 3. The 28 students in Mr. Tabor’s AP Statistics class completed a brief survey. One of the questions asked whether each student was right- or left-handed. The two-way table summarizes the class data. Choose a student from the class at random. The events of interest are “female” and “right-handed.”   Gender Handedness Female Male Left 3 1 Right 18 6 Independent. Once we know that the chosen person is female, this does not tell use anything more about whether she is right-handed. Students who are right handed P(A) = 24/28 = 6/7 and the P(A│B) = 18/21 = 6/7

Independent vs. Disjoint/Mutually Exclusive A. Disjoint Events – two events that cannot happen at the same time (no outcomes in common!) EX: A-Green light, B-red light, C-yellow light EX: A-Junior, B-Senior B. Independent Events – one does not change the probability of another EX: A-wear black socks B-likes pizza EX: A-car you drive B-eye color

Examples If two events are disjoint, they will NEVER be independent! (Because if one happens, the other one is guaranteed NOT to happen) Ex: A – roll an even # B – roll an odd # If two events are not disjoint: They may be independent. A – like pizza B – made an A A – female B – junior They may be not independent. A – baseball B -- rain

More examples Are red card and spade independent? Mutually exclusive? There’s a 25% chance you have a spade, but if you know you have a red card you definitely DON’T have a spade. NOT INDEPENDENT There are no red cards that are spades, so the two events are MUTUALLY EXCLUSIVE.

Are red card and ace independent. Mutually Exclusive Are red card and ace independent? Mutually Exclusive? The probability of getting a red card is 0.5. The probability the card is chosen from the four aces is also 0.5. P(red) = P(red | ace) INDEPENDENT You can have a red ace. They are NOT MUTUALLY EXCLUSIVE.

Are “face card” and “king” independent? Mutually Exclusive? “Face card” and “king” are not independent. The probability of drawing a face card is 12/52. The probability of drawing a face card from the 4 kings is 1. NOT INDEPENDENT P(face card) ≠ P(face card | king) You can have a face card that is a king. NOT MUTUALLY EXCLUSIVE.

Tree Diagrams Another way to model chance behavior that involves a sequence of outcomes is called a tree diagram.    A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree. Each branch should sum to 1. The far right represents all possible outcomes and they should also sum to 1. To get the probability of a given outcome, follow along the “branches” and multiply the probabilities of each. This strategy will also work even when trials are not independent.

Tree Diagrams What is the probability that two students selected at random will both be left-handed?     Gender Handedness Female Male Total Left 3 1 4 Right 18 6 24 21 7 28

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) This rule says that for both events to occur, first one must occur, and then given that the first event has occurred, the second must occur.

Tree Diagrams & General Mult. Rule 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? Explain why your answer makes sense. (HINT: Draw a Tree Diagram.) 

Tree Diagrams & General Mult. Rule In a certain city, 45% of registered voters are Republicans, 40% are Democrats, and the rest are Independents. Of those, 40% of the Republicans are women, 55% of the Democrats are women, and 60% of the Independents are women. Express the information in a tree diagram Suppose you randomly select a registered voter: What is the probability that you choose a person who is Republican and a woman? What is the probability that you choose a person who is a woman? What is the probability that you choose a person who is male or an Independent? 

Tree Diagrams & General Mult. Rule Video-sharing sites, led by YouTube, are popular destinations on the Internet. Let’s look only at adult Internet users, aged 18 and over. About 27% of adult Internet users are 18-29 years old, another 45% are 30-49 years old, and the remaining 28% are 50 and over. The Pew Internet and American Life Project finds that 70% of Internet users aged 18-29 have visited a video-sharing site, along with 51% of those aged 30-49 and 26% of those 50 and older. Do most Internet users visit YouTube and similar sites? What percent of all adult Internet users visit video-sharing sites? (Use a Tree Diagram)

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). Derive this rule from the General Multiplication Rule: P(A∩B) = P(A) ● P(B|A) General Multiplication Rule

Multiplication Rule for Independent Events On January 28, 1986, Space Shuttle Challenger exploded on takeoff. All seven crew members were killed. Following the disaster, scientists and statisticians helped analyze what went wrong. They determined that the failure of O-ring joints in the shuttle’s booster rockets was to blame. Under the cold conditions that day, experts estimated that the probability that an individual O-ring joint would function properly was 0.977. But there were six of these O-ring joints, and all six had to function properly for the shuttle to launch safely. Assuming that O-ring joints succeed or fail independently, find the probability that the shuttle would launch safely under similar conditions.

Probability of At Least One Sometimes it really only matters if something occurs once. Examples include floods, hurricanes, natural disasters. Suppose the probability of an event A occurring in one trial is P(A). If all trials are independent, the probability that event A occurs at least once in n trials is the complement of the probability of the event never occurring. Therefore, the probability is: P(at least 1) = 1 – P(event not occurring at all)

Probability of At Least One What is the probability that a region will experience at least one 100-year flood (a flood that has a 0.01 chance of occurring in any given year) during the next 100 years? Assume that 100-year floods in consecutive years are independent events. If there is a 0.01 chance of the event occurring, then there’s a 0.99 chance that it won’t occur—the complement. Thus, for there to be AT LEAST ONE flood in the next 100 years, you calculate the complement

Probability of At Least One If there are 30 students in a class, what is the probability that at least one person in the class has the same birthday as me?

Probability of At Least One Many people who come to clinics to be tested for HIV, the virus that causes AIDS, don’t come back to learn the test results. Clinics now use “rapid HIV tests” that give a result while the client waits. In a clinic in Malawi, for example, use of rapid tests increased the percent of clients who learned their test results from 69% o 99.7%. The trade-off for fast results is that rapid tests are less accurate than slower laboratory tests. Applied to people who have no HIV antibodies, one rapid test has probability of about 0.004 of producing a false positive (that is, of falsely indicating that antibodies are present). If a clinic tests 200 people who are free of HIV antibodies, what is the chance that at least one false positive will occur?

Calculating Conditional Probabilities Rearrange the General Multiplication Rule to find and equation for P(B|A). P(B|A) = This is called the Conditional Probability Formula.

Calculating Conditional Probabilities Given the Venn Diagram below and the events A: reads USA Today B: reads NY Times What is the probability that a randomly selected resident who reads USA Today also reads the NY Times? A B 35% 5% 20% 40%

Calculating Conditional Probabilities Athletes and Drug Testing: Over 10,000 athletes competed in the 2008 Olympic Games in Beijing. The International Olympic Committee wanted to ensure that the competition was as fair as possible. So the committee administered more than 5000 drug tests to athletes. All medal winners were tested, as well as other randomly selected competitors. Suppose that 2% of the athletes had actually taken (banned) drugs. No drug test is perfect. Sometimes the test says that an athlete took drugs, but the athlete actually didn’t. We call this a false positive result. Other times, the drug test says an athlete is “clean,” but the athlete actually took the drugs. This is called a false negative result. Suppose that the testing procedure use at the Olympics has a false positive rate of 1% and a false negative rate of 0.5%. What is the probability that an athlete who tests positive actually took drugs?