Section 4.2 Some Probability Rules—Compound Events

2 P(Event A AND Event B) -Multiply probabilities -BUT first consider if events are independent or not -Ex 1: Roll two fair die. P(rolling a 5 on both) = -Ex 2: Six marbles in a bag (3 green, 2 blue, 1 red) P(2 green balls w/replacement) = P(2 green balls w/o replacement) =

3 cont’d For ex.1

4 Conditional Probability Notation: P(A|B) means given event B occurred, it’s the probability of event A occurring OR P(B|A) means given event A occurred, it’s the probability of event B occurring

5 Ex 3 P(A= Andrew will be alive in 10 yrs) = 0.72 P(B= Ellen will be alive in 10 yrs) = 0.92 Assuming their lives don’t effect each other… P(both will be alive in 10 yrs) =

6 Ex digital cameras. Drawing 2 at random to check quality (w/o replacement). The lot contains 10 defective cameras. P(both cameras drawn are defective) =

7 Multiplication Rule if events are Independent Multiplication Rule if events are not Independent

8 Notes: If two events are independent then, P(A | B) = P(A) Or P(B|A) = P(B) Ex: P(rolling a 3, given you rolled a 4) = And you can solve for the conditional probability:

9 P(Event A OR Event B) -Add probabilities -BUT consider if events are disjoint (or mutually exclusive) -Disjoint Events: are events that cannot occur together. So P(A and B) = 0

10 Ex 5 31 students total: 15 freshmen (9 girls, 6 boys) 8 sophomores (3 girls, 5 boys) 6 juniors (4 girls, 2 boys) 2 seniors (1 girl, 1 boy) a) P(randomly selecting a freshman or sophomore) =

11 (b) P(selecting a male or a sophomore) = cont’d

12 Addition Rule for disjoint events Addition Rule for non-disjoint events

13 Ex 6 P(slacks being too tight) = 0.30 P(slacks being too loose) = 0.10 a)Are the events mutually exclusive? b)P(too tight or too loose) =

14 Ex 7 Professor is preparing an exam. P(students need work in math) = 0.80 P(students need work in english) = 0.70 P(need both areas) = 0.55 a)Are the events mutually exclusive? b)P(need math or need english) =

15 Contingency Tables Employee Type Democrat (D)Republican (R) Independent (I) Row Total Executive (E) Production worker (PW) Column total (grant total) a)P(D) =P(E) = b)P(D | E) = c)Are D and E independent? d)P(D and E) = e) P(D or E) =

16 Contingency Table continued… a)P(I) =P(PW) = b)P(I | PW) = c)P(I and PW) = d)Use the multiplication rule for P(I and PW) = e)Is the answer in c the same as d? f)P(I or PW) g)Are I and PW mutually exclusive? Employee Type Democrat (D)Republican (R) Independent (I) Row Total Executive (E) Production worker (PW) Column total (grant total)

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