Chapter 15 B: The Multiplication Rule & Conditional Probabilities

Objective: To use the addition rule to calculate probabilities

CHS Statistics

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

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?

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

1. A coin is tossed and a die is rolled. Find the probability of getting a heads and then rolling a 6.

2. Consider tossing a coin twice. What is the probability of landing on heads twice?

 

Examples: Multiplication Rule

3. 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 Rule4. 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?

5. 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

a) with replacement.

b) without replacement.

Examples: Multiplication Rule6. Find the probability of Event A occurs given that Event B

already occurred.

a) P(2 spades) =

b) P(Even number on a die, given that the result of the die is 3 or less) =

c) P(Heart | Red) =

 

7. Find the probability of a couple having at least 1 girl among 3 children.

•  

Examples: Contingency TablesThe table below shows the results of a study where researchers examined a child’s IQ and the presence of a specific gene.  

a) Find the probability the child has a Normal IQ.

b) Find the probability that a child has a high IQ, given that the child has the gene.

Examples: Contingency TablesThe table below shows the results of a study where researchers examined a child’s IQ and the presence of a specific gene.  

c) Find the probability that a child does not have the gene and has high IQ. 

d) Find the probability that a child does not have the gene, given that the child has a normal IQ.

e) 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!