CHAPTER 5 Probability: What Are the Chances? 5.3 Conditional Probability and Independence
Conditional Probability and Independence 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.
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). Read | as “given that” or “under the condition that”
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 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. Total 3392 2952 3656 10000 Total 6300 1600 2100 Find P(L) Find P(E | L) Find P(L | E) Try Exercise 39 P(L) = 3656 / 10000 = 0.3656 P(E | L) = 800 / 3656 = 0.2188 P(L| E) = 800 / 1600 = 0.5000
Joint Probability A B S
Conditional Probability Since Event A has happened, the sample space is reduced to the outcomes in A A P(A and B) represents the outcomes from B that are included in A S
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. 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.
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
Example: Tree Diagrams 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? 51.15% of teens are online and have posted a profile.
3. In Mr. Jonas' homeroom, 70% of the students have only brown hair, 25% have only brown eyes, and 5% have both brown hair and brown eyes. A student is excused early to go to a doctor's appointment. If the student has brown hair, what is the probability that the student also has brown eyes? P(brown eyesbrown hair) = P(brown eyes and brown hair)/P(brown hair) = .05/.7 = .071 The probability of a student having brown eyes given he or she has brown hair is 7.1%
4. Suppose we survey all the students at school and ask them how they get to school and also what grade they are in. The chart below gives the results. Complete the two-way frequency table: Bus Walk Car Other Total 9th or 10th 106 30 70 4 11th or 12th 41 58 184 7
Suppose we randomly select one student. Bus Walk Car Other Total 9th or 10th 106 30 70 4 210 11th or 12th 41 58 184 7 290 147 88 254 11 500 Suppose we randomly select one student. a. What is the probability that the student walked to school? 88/500 17.6% b. P(9th or 10th grader) 210/500 42% c. P(rode the bus OR 11th or 12th grader) 147/500 + 290/500 – 41/500 396/500 or 79.2%
Bus Walk Car Other Total 9th or 10th 106 30 70 4 210 11th or 12th 41 58 184 7 290 147 88 254 11 500 d. What is the probability that a student is in 11th or 12th grade given that they rode in a car to school? P(11th or 12thcar) * We only want to look at the car column for this probability! = 11th or 12th graders in cars/total in cars = 184/254 or 72.4% The probability that a person is in 11th or 12th grade given that they rode in a car is 72.4%
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). 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 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 0.977. 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).
Conditional Probabilities and Independence 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.