1. AP Statistics Probability Rules

2. Definitions Probability of an Outcome: A number that represents the likelihood of the occurrence of an outcome. Probability of an Outcome: A number that represents the likelihood of the occurrence of an outcome. Alt.: The proportion of times the outcome would occur in a very long series of repetitions. Alt.: The proportion of times the outcome would occur in a very long series of repetitions.

3. Definitions Theoretical Probability: probability values arrived at through calculations. Theoretical Probability: probability values arrived at through calculations. Empirical Probability: probability values arrived at through observation or simulation. Empirical Probability: probability values arrived at through observation or simulation.

4. Basic Rules for Probability All possible outcomes must have a combined probability of 1.

5. Basic Rules for Probability P(A) = 0 if and only if A is certain not to occur (impossible event). P(A) = 0 if and only if A is certain not to occur (impossible event). P(A) = 1 if and only if A is certain to occur. P(A) = 1 if and only if A is certain to occur. The probability that event A will fail to occur is denoted P(A c ), or the complement of A. The probability that event A will fail to occur is denoted P(A c ), or the complement of A. P(A c ) = 1 – P(A). P(A c ) = 1 – P(A).

6. The Complement Rule  The Complement Rule states that; P(A)+P(A c )=1  P(Sample Space) = 1, because the sample space represents all possible outcomes. Therefore:  P(A)+P(A c ) = P(S) = 1  Either event A happens or it doesn’t happen. There are no other choices!  “At Least One” Rule  P(At least one) = 1 – P(none)

7. Another Probability Rule Another Probability Rule: If A and B are disjoint (can’t happen at the same time), then: Another Probability Rule: If A and B are disjoint (can’t happen at the same time), then: P(A or B) = P(A) + P(B) This is the Addition Rule for Disjoint Events. This is the Addition Rule for Disjoint Events.

8. Addition Rule The word “or” means “addition” in the language of probability. The word “or” means “addition” in the language of probability. Note that P(A or B) can also be written as: Note that P(A or B) can also be written as: and can be stated as “A union B”.

9. Example For women age 25 to 29: For women age 25 to 29: P(never married or divorced) = P(never married) + P(Divorced) P(never married or divorced) = P(never married) + P(Divorced) =.353 +.071 =.424 =.353 +.071 =.424 Marital Status Never Married MarriedWidowedDivorced Probability.353.574.002.071

10. Independence Definition: Two events are independent when knowing that one occurred does not change the probability that the other occurred. Definition: Two events are independent when knowing that one occurred does not change the probability that the other occurred. Coin flips and dice rolls are independent. Coin flips and dice rolls are independent.

11. Independence If two events are independent, then the probability of both A and B occurring is: If two events are independent, then the probability of both A and B occurring is: P(A and B) = P(A)∙P(B) P(A and B) = P(A)∙P(B) This is the Multiplication Rule for Independent Events. This is the Multiplication Rule for Independent Events. Example: we roll a die twice with P(even)=.5 and P(odd) =.5. Example: we roll a die twice with P(even)=.5 and P(odd) =.5. P(even and odd) = (.5)(.5) =.25 P(even and odd) = (.5)(.5) =.25

12. Multiplication Rule for Independent Events The word “and” means “Multiplication” in the language of probability. The word “and” means “Multiplication” in the language of probability. Note that P(A and B) can also be written as: Note that P(A and B) can also be written as: and can be stated as “A intersect B”.

13. Example For Dependent Events

14. Homework Worksheet. Worksheet.