Chapter 15 B: The Multiplication Rule & Conditional Probabilities CHS Statistics Chapter 15 B: The Multiplication Rule & Conditional Probabilities Objective: To use the addition rule to calculate probabilities

Warm-up: Something to Consider… Consider the following two test questions: 1) True or False: Ms. Halliday’s favorite color is orange.   2) Ms. Halliday’s favorite sports team is: a) Pittsburgh Steelers b) Pittsburgh Penguins c) Baltimore Ravens d) New England Patriots e) Pittsburgh Pirates If you do not know and you guess, what is the probability that you answer both questions correctly?

Independent vs. Dependent Events Independent Events: when the outcome of one event does not affect the probability of the other event Examples: Rolling a 3 on a die, then rolling a 4 Flipping a coin and getting heads, then flipping a coin again.   Dependent Events: when the outcome of one event affects the probability of the other event   Example: In the envelope activity, a student selected an envelope, and it was not replaced. The probabilities of the other events changed.

Examples: Dependent vs. Independent Decide whether the following events are independent or dependent: Tossing a coin and getting a heads and then rolling a six-sided die and getting a 6 Eating 10 cheeseburgers in a row and then getting a stomach ache

Experiment: Conditional Probability Experiment: Toss a coin. If it lands on heads, you draw from Bag 1. Bag 1 contains 2 green marbles and 1 blue marble. If it lands on tails, you pick from Bag 2. Bag 2 has 1 green marble and 3 blue marbles.   P(blue given tails was flipped)=

Conditional Probability The probability of Event B occurring after it is assumed the Event A has already occurred. P(B|A) is read as the probability of B given A.

Conditional Probability Rule 𝑷 𝑩 𝑨 = 𝑷(𝑨 𝒂𝒏𝒅 𝑩) 𝑷(𝑨) One occurrence of both A and B – not the multiplication rule that requires two or more events/occurrences

Independence in Terms of Conditional Probabilities Independence of two events means that the outcome of one event does not influence the probability of the other. With our new notation for conditional probabilities, we can now formalize this definition: Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)

Independence Example 37% of working adults access the Internet at work. 44% access the Internet from home, and 21% access the Internet at both work and home. Are accessing the internet from work and accessing the internet from work independent? Are they mutually exclusive?

P(A and B) = P(A) ∙ P(B|A) Multiplication Rule P(A and B) = P(A) ∙ P(B|A) The probability of Event A times the Probability of Event B occurring, given Event A already occurred. If your events are INDEPENDENT, your second probability won’t be affected by the first, so you would just multiply the two probabilities together. If your events are DEPENDENT, you have to calculated the second, given that the first already occurred.

Examples: Multiplication Rule A coin is tossed and a die is rolled. Find the probability of getting a heads and then rolling a 6. Consider tossing a coin twice. What is the probability of landing on heads twice?  

Examples: Multiplication Rule The probability that a particular knee surgery is successful is 0.85. Find the probability that three knee surgeries will be successful. Find the probability that none of three knee surgeries is successful.

Examples: Multiplication Rule A bag contains 2 red cubes, 3 blue cubes, and 5 green cubes. If a red cube is removed, what is the probability that a green cube will be picked? A pool of potential jurors consists of 10 men and 15 women. The Commissioner of Jurors randomly selects two names from this pool. Find the probability that the first is a man and the second is a man if two people are selected with replacement. without replacement.

Examples: Multiplication Rule Find the probability of Event A occurs given that Event B already occurred. P(2 spades) = P(Even number on a die, given that the result of the die is 3 or less) = P(Heart | Red) =   Find the probability of a couple having at least 1 girl among 3 children.

Examples: Contingency Tables The table below shows the results of a study where researchers examined a child’s IQ and the presence of a specific gene.   Find the probability the child has a Normal IQ. Find the probability that a child has a high IQ, given that the child has the gene.

Examples: Contingency Tables The table below shows the results of a study where researchers examined a child’s IQ and the presence of a specific gene.   Find the probability that a child does not have the gene and has high IQ.  Find the probability that a child does not have the gene, given that the child has a normal IQ. Find the probability that a child has Normal IQ or Gene Present.

Assignment pp. 361-365 # 6 – 12 Even, 13 – 19 Odd, 23 - Check your solutions online!