chapter 15 probability rules!. the general addition rule does not require disjoint events! p(a u b)...
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EXAMPLE: USING THE GENERAL ADDITION RULE A survey of college students found that 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. What is the probability that a randomly selected student either lives OR eats on campus?TRANSCRIPT
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C H A P T E R 1 5
PROBABILITY RULES!
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THE GENERAL ADDITION RULE
• Does NOT require disjoint events!
• P(A U B) = P(A) + P(B) – P(A ∩ B)
• Add the probabilities of A and B then subtract the probability of A and B.
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EXAMPLE:USING THE GENERAL ADDITION RULE
A survey of college students found that 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. What is the probability that a randomly selected student either lives OR eats on campus?
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EXAMPLE:USING VENN DIAGRAMS
Back to the college students: 56% live on campus, 62% have a meal plan, and 42% do both. Based on a Venn diagram, what is the probability that a randomly selected studenta) Lives off campus and doesn’t have a meal plan?b) Lives on campus but doesn’t have a meal plan?
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STEP-BY-STEP EXAMPLE
Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests.
What is probability that a randomly selected DWI suspect is given…1. A test?2. A blood test or a breath test but not both?3. Neither test?
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CONDITIONAL PROBABILITY: IT DEPENDS…
Conditional Probability: a probability that takes into account a given condition
P(B│A) = the probability of B given that A has occurred
**A usually tells you what to put in the denominator
P(B│A) =
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FINDING CONDITIONAL PROBABILITYEXAMPLE
Our survey found that 56% of college students live on campus, 62% have a campus meal program, and 42% do both. While dining in a campus facility open only to students with meal plans, you meet someone interesting. What is the probability that your new acquaintance lives on campus?
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INDEPENDENCE
Events A and B are independent whenever P(B│A) = P(B).
Are living on campus and having a meal plan independent? Are they disjoint? (Back it up with math)
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INDEPENDENT ≠ DISJOINT
• Disjoint events can NOT be independent!
• Two events could be either independent or disjoint, but not both.
• And they can be NEITHER disjoint nor independent.
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TABLES VS VENN DIAGRAMS
• Step-by-Step Example
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THE GENERAL MULTIPLICATION RULE
• Does not require independence!
• P(A∩B) = P(A) x P(B│A)
• Means the probability of A and B equals the probability of A times the probability of B given A has occurred.
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EXAMPLE:USING THE GENERAL MULTIPLICATION RULE
A factory produces two types of batteries, regular and rechargeable. Quality inspection tests show that 2% of the regular batteries come off the manufacturing line with a defect while only 1% of the rechargeable batteries have a defect. Rechargeable batteries make up 25% of the company’s production. What is the probability that if we choose 1 battery at random we get…a) A defective rechargeable battery?b) A regular battery and it is not defective?
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DRAWING WITHOUT REPLACEMENT
You just bought a small bag of Skittles. Not that you could know this, but inside are 20 candies: 7 green, 5 orange, 4 red, 3 yellow, and only 1 purple. You tear open one corner of the package and begin eating them by shaking one out at a time. What is the probability that …A) Your first two Skittles are both orange?
B) That none of your first 3 candies is green?
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TREE DIAGRAMS
• A display of conditional events or probabilities that is helpful in organizing our thinking.
• Now lets make a tree diagram with the battery example…