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Chapter 15 B: The Multiplication Rule & Conditional Probabilities
Objective: To use the addition rule to calculate probabilities
CHS Statistics
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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?
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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.
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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
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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)=
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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.
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Conditional Probability Rule
One occurrence of both A and B – not the multiplication rule that requires two or more events/occurrences
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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).)
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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?
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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.
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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?
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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.
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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.
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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.
•
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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.
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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.
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Assignment
pp. 361-365 # 6 – 12 Even, 13 – 19 Odd, 23
- Check your solutions online!