AP Statistics

1. AP Statistics Chapter 6 Probability and Simulation

2. Section 6.1 - Simulation • In this section, we investigate the building of simulations to answer various questions involving chance. • Simulation: • The imitation of chance behavior, based on a model that accurately reflects the phenomenon under consideration • On the AP exam, students may receive full credit for solving a problem by simulation, even if the problem does not call for it. • You can use a coin, dice, calculators, computers, or random number tables

3. Section 6.1 - Simulation • Simulation Steps: • State the problem or describe the random phenomenon. • What are you simulating? • State the assumptions that you are making. • How are you simulating (what are you using)? • Independent assumption • Assign digits to represent outcomes • Simulate many repetitions (trials) • State your conclusions

5. Probability Terms • Random: Individual outcomes are uncertain, but there is a predictable distribution of outcomes in the long run. • Probability: long term relative frequency • Sample Space: The set of all possible outcomes of a random phenomenon. • Sample space for rolling one die • S = {1, 2, 3, 4, 5, 6} • Sample space for the heights of adult males • S = {all real x such that 30in < x < 100in}

6. Ways to determine Sample Space • 1. Tree diagram • 2. Multiplication Principle: If one task can be done n1 number of ways and another can be done n2 number of ways, then both tasks can be done in n1 × n2 number of ways. • 3. Organized list

7. Events • Any outcome or set of outcomes of a random phenomenon. (It is a subset of the sample space). • Ex: rolling a 1 • Ex: rolling a 2 or 3 • Ex: Randomly choosing an adult male between 60 and 65 inches tall.

8. Other probability terms • Sampling with replacement: Each pick is the same…(number goes back in the hat). • Sampling without replacement: Each draw is different. • Mutually exclusive/disjoint: Two (or more) events have no outcomes in common and thus can never occur simultaneously. • Complement: The complement of any event, A, is the event that A does not occur. (Ac)

9. Basic Probability Rules • 1. For any event, A, 0 < P(A) < 1. • 2. If S is the sample space, then P(S) = 1. • 3. Addition Rule: If A and B are disjoint, then • P(A or B) = P(A U B) = P(A) + P(B) • 4. Complement Rule: P(Ac) = 1 – P(A)

10. Set Notation

11. More Set Notation

12. Independence • Independence: Knowing that one event occurs does not change the probability that the other event occurs. • 5. Multiplication Rule • If events A and B are independent, then • P(A and B) = P(A ∩ B) = P(A) × P(B)

14. General Addition Rule • Reminder….addition rule for mutually exclusive events is… • P(A U B U C….) = P(A) + P(B) + P(C) + … • The General Addition Rule applies to the union of two events, disjoint or not. • P(A or B) = P(A) + P(B) – P(A and B) • P(A U B) = P(A) + P(B) – P(A ∩ B)

15. Conditional Probability • P(A|B)  “The probability of event A given that event B has occurred.” • Examples: • One card has been picked from a deck. Find… • P(spade|black), P(queen|face card) • One dice has been rolled. Find… • P(3|odd), P(odd|prime) • Two dice are rolled. Find P(2nd die is 4|1st die is 3). • New definition of independence: Events A and B are independent if P(A) = P(A|B).

16. General Multiplication Rule • Reminder….Multiplication Rule for independent events is… • P(A ∩ B) = P(A) × P(B) • The General Multiplication Rule applies to the intersection of two events, independent or not. • P(A ∩ B) = P(A) × P(B|A) • P(A ∩ B) = P(B) × P(A|B) • Why does this rule also work for independent events?

17. Conditional Probability Formula