Report

AP STATISTICS Types of Probability Theoretical: true mathematical probability Empirical: the relative frequency with which an event occurs in a given experiment Subjective: an educated guess Review Terms Experiment: any process that yields a result or observation Outcome: a particular result of an experiment Sample Space: the collection of all possible outcomes Event: any collection of outcomes; any subset of the sample space Example: Roll die Let A={1} Let B={2, 4, 6} Let C={3, 6} An event occurs if any outcome of the event occurs Probability of Event P(A) = # of outcomes/# outcomes in sample space If each outcome is equally likely Using previous slide’s events: P(A)= P(B)= P(C)= Rules of Probability Rule 1: For any event, 0 ≤ () ≤ 1 Rule 2: Complement Rule “A complement” is the event that A does not occur; the set of all outcomes not in A + ′ = 1 Example: even/odd; alive/dead P(ACE)= P(ACE’)= Rules of Probability Rule 3: Mutually Exclusive/disjoint-two events, A & B that have no outcomes in common Examples: red and spade; freshman, sophomore, junior Note: complementary→mutually exclusive ; mutually exclusive→complementary Rule 4: Additive Rule for disjoint events ∪ = + 1 2 1 4 Example: P(red or spade)=P(red) + P(spade)= + = 3 4 Rules of Probability Rule 5: Additive Rule-For any events A & B, P(A or B)=P(A)+P(B)-P(A and B) Example: Find the probability of drawing red card or Ace. ∪ = + − = 26 4 2 + − 52 52 52 = 28 52 Rule 6: Conditional Probability-For two events A and B, the probability that A occurs given that B has occurred. = (∩) () Probability Rules = (∩) () Example: What is the probability that a card is a diamond, given that it is red? ( ∩ ) 13/52 13 = = = = .5 () 26/52 26 Example: What is the probability of 2, given that you got an even number? (2 ∩ ) 1/6 2 = = = 1/3 () 3/6 Independent Events 2 events A and B are independent if the occurrence of one event doesn’t affect the probability of occurrence of the other event To prove: = = Example: rolling two dice; drawing from a deck with replacement Are drawing a face card and drawing a red card independent events? = ( ∩ ) = 12/52 () 6/52 = 12/52 26/52 So they are independent. Probability Rules Rule 7: Multiplicative Rule for Independent Events ∩ = () Has to be shown or given Example: Find the probability of drawing two Queens from a deck of cards if it is done with replacement. 4 4 1 ∩ 2 = 1 2 = ∗ ≈ .0059 52 52 Rule 8: Multiplicative Rule- For any two events A& B, the ∩ = or = () Example: Find the probability of drawing two Queens from a deck of cards if it is done without replacement 4 3 1 ∩ 2 = 1 2 1 = ∗ ≈ .0045 52 51 Example: Contingency Table G PG PG-13 R 2000s 2 13 22 2 1990s 1 1 7 0 1980s 0 1 0 0 1970s 0 1 0 0 9 P(1990’s)= 50 29 P(PG-13)= 50 = .18 = .58 P(1990s and PG-13)= 7 50 = .14 Are 1990s and PG-13 disjoint events? G PG PG-13 R 2000s 2 13 22 2 1990s 1 1 7 0 1980s 0 1 0 0 1970s 0 1 0 0 The probability that a randomly selected DVD is rated PG-13 or is from the 1990s. P(PG-13 or 1990s)=P(PG13)+P(90s)-P(PG13 and 90s) =.58+.18-.14=.62 90 13 = (90 ∩13) (13) = 7/50 29/50 = 7/29 Are PG-13 and 1990s independent or dependent events? 90 13 ≠ (90) dependent Other ways: 13 = 13 90 13 90 = 13 (90) Homework 6.69, 71, 78 Have a wonderful weekend!