S ECTION 7.2: P ROBABILITY M ODELS

P ROBABILITY M ODELS A Probability Model describes all the possible outcomes and says how to assign probabilities to any collection of outcomes (events). Example: All possible outcomes listed. All probabilities assigned total 1. Marital Status Never Married MarriedWidowedDivorced Probability

R ULES OF P ROBABILITY Any probability is a number between 0 and 1. An event with probability 0 never occurs. An event with probability 1 occurs on every trial. All probabilities must add up to a total of 1. The probability that an event does not happen is 1 minus the probability that the event does happen. P(Eʹ) = 1 – P(E) Eʹ is called E prime and is the “complement” of E. E is the event you are checking the probability of. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities (Addition Rule).

C ONDITIONAL P ROBABILITY Conditional probability is the probability of an event occurring, given that another event has already occurred. The conditional probability of Event B occurring, given that Event A has occurred is denoted by P(B|A) and is read as “Probability of B, given A”. Example: Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced)…

A NSWER … Because the first card is a king and has not been replaced, there are now only 51 cards. So the probability that the second card chosen is a queen will be 4/51 or

C ONDITIONAL P ROBABILITY Researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given the child has the gene. P(A|B) = 33/72 = Gene Present Gene Not Present Total High IQ Normal IQ Total

I NDEPENDENT OR D EPENDENT ? Two events are independent if the occurrence of one does not affect the occurrence of the other (mutually exclusive)… P(B|A) = P(B) or if P(A|B) = P(A) Events which are not independent are dependent. Ex., Selecting a king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B). P(B|A) = 4/51 P(A|B) = 4/52 The occurrence of A changes the probability of the occurrence of B, so the events are dependent. If the card was replaced, is it still dependent?

M ULTIPLICATION R ULE The probability that two events A and B will occur in sequence is… P(A and B) = P(A) * P(B|A) If events A and B are independent, then the rule can be simplified to… P(A and B) = P(A) * P(B) This can be used for any number of events.

M ULTIPLICATION R ULE Two cards are selected from a standard deck without replacement. Find the probability that both are hearts. P(H1 and H2) = P(H1) * P(H2|H1) =13/52 * 12/51 = 156/2652 = A coin is tossed and a die is rolled. Find the probability of getting a tail and rolling a 6. P(T and 6) = P(T) * P(6) =1/2 * 1/6 = 1/12 = 0.083

E XAMPLE In a box of 11 parts, four of the parts are defective. Two parts are selected without replacement. Find the probability that both parts are defective. 4/11 * 3/10 = 12/110 = Find the probability that both parts are not defective. 7/11 * 6/10 = 42/110 = Find the probability that at least one is defective. This is the complement of E…1 – 4/11 = 7/11 = 0.636

W ORK … Group Work: White Book: Page , #11, 12, Homework: Green Book: Pg , #19-23 Green Book: Pg. 439, #33 Green Book: Pg. 443, #43