1. 3.2-Conditional Probability The probability of an event occurring given another event has already occurred. P(B|A) = “Probability of B, given A” # outcomes in event / # outcomes in sample space. B/A NO REPLACEMENTS

2. Examples 2 cards are selected WITHOUT replacement. What is the probability the second is a queen given the first is a king? From table on p. 115, what is probability the child has a high IQ given it has the gene? Do the TRY IT YOURSELF 1 on p. 115

3. Examples 2 cards are selected WITHOUT replacement. What is the probability the second is a queen given the first is a king? 4 queens, 51 cards left so 4/51 = 0.078 From table on p. 115, what is probability the child has a high IQ given it has the gene? Do the TRY IT YOURSELF 1 on p. 115

4. Examples 2 cards are selected WITHOUT replacement. What is the probability the second is a queen given the first is a king? 4 queens, 51 cards left so 4/51 = 0.078 From table on p. 115, what is probability the child has a high IQ given it has the gene? 33 high IQ with gene out of 72 with gene so 33/72 = 0.458 Do the TRY IT YOURSELF 1 on p. 115

5. TRY IT YOURSELF 1 1a) # of outcomes of event (no gene) = 30 # of outcomes of ss (total kids)= 102 b) P(no gene) = 30/102 = 0.294 2 a) # of outcomes of event ( no gene normal IQ) = 11 # of outcomes of ss (total with normal IQ) = 50 b) P(no gene|normal IQ) = 11/50 = 0.22

6. Independent & Dependent Events Independent Events – Occurrence of one event does NOT affect the other – P(B|A) = P(B) OR P(A|B)=P(A) Dependent Events – Occurrence of one event DOES affect the other – Non-replacing – Sample space changes each time

7. Examples: Independent or Dependent? What is the probability? Selecting a king and then a queen (no replacement)? Tossing a coin heads, then rolling a 6 on a 6 sided die? Practicing the piano and then becoming a concert pianist? Do TRY IT YOURSELF 2 p. 116

8. Examples: Independent or Dependent? What is the probability? Selecting a king and then a queen (no replacement)? Dependent P(B|A) = 4/51, P(B) = 4/52 )not same Tossing a coin heads, then rolling a 6 on a 6 sided die? Practicing the piano and then becoming a concert pianist? Do TRY IT YOURSELF 2 p. 116

9. Examples: Independent or Dependent? What is the probability? Selecting a king and then a queen (no replacement)? Dependent P(B|A) = 4/51, P(B) = 4/52 )not same Tossing a coin heads, then rolling a 6 on a 6 sided die? Independent : P(B|A)=1/6, P(B) = 1/6 same Practicing the piano and then becoming a concert pianist? Do TRY IT YOURSELF 2 p. 116

10. Examples: Independent or Dependent? What is the probability? Selecting a king and then a queen (no replacement)? Dependent P(B|A) = 4/51, P(B) = 4/52 )not same Tossing a coin heads, then rolling a 6 on a 6 sided die? Independent : P(B|A)=1/6, P(B) = 1/6 same Practicing the piano and then becoming a concert pianist? Dependent: practicing affects chances of it Do TRY IT YOURSELF 2 p. 116

11. TRY IT YOURSELF 2 1. – A) No – B)Independent – C) making it through first has no affect on second 2. – A) Yes – B) Dependent – C) Studies show exercise lowers resting heart rate

12. Multiplication Rule: P(A AND B) The probability that 2 events A and B will occur in sequence is: Dependent: P(A and B) = P(A) · P(B|A) Independent: P(A and B) = P(A) · P(B) AND Can be extended for any number of events IF P(B) = P(B|A), then A and B are independent and simpler rule of multiplication can be used.

13. Examples: 2 cards are selected without replacement. What is the probability of a king AND then a queen? A coin is tossed AND a die is rolled. What is the probability of getting a head AND rolling a 6? Do TRY IT YOURSELF 3 p. 117

14. Examples: 2 cards are selected without replacement. What is the probability of a king AND then a queen? dependent P(K and Q)=P(K)·P(Q|K)=4/52 ·4/51=0.006 A coin is tossed AND a die is rolled. What is the probability of getting a head AND rolling a 6? Do TRY IT YOURSELF 3 p. 117

15. Examples: 2 cards are selected without replacement. What is the probability of a king AND then a queen? dependent P(K and Q)=P(K)·P(Q|K)=4/52 ·4/51=0.006 A coin is tossed AND a die is rolled. What is the probability of getting a head AND rolling a 6? independent P(H and 6) = P(H)·P(6)=1/2 · 1/6 = 1/12=.083 Do TRY IT YOURSELF 3 p. 117

16. TRY IT YOURSELF 3 1. A = swim thru first B= swim thru 2nd – A) independent – B) P(A and B) = P(A)·P(B) = (0.85)(0.85) = 0.7225 2. A=no gene B=normal IQ – A) Dependent – B) P(A and B) = P(A)·P(B|A) = 30/102·11/30 =.108

17. Examples: Find the probabilities A coin is tossed AND a die is rolled. What is the probability of a head AND then a 2? Probability of 1 salmon getting through a dam is 0.85. What is the probability of 3 getting through the dam? Probability that none of the salmon get through? Probability that at least one gets through? Do TRY IT YOURSELF 4 on p. 118

18. Examples: Find the probabilities A coin is tossed AND a die is rolled. What is the probability of a head AND then a 2? independent: P(A)·P(B)=1/2·1/6 = 1/12=0.083 Probability of 1 salmon getting through a dam is 0.85. What is the probability of 3 getting through the dam? Probability that none of the salmon get through? Probability that at least one gets through? Do TRY IT YOURSELF 4 on p. 118

19. Examples: Find the probabilities A coin is tossed AND a die is rolled. What is the probability of a head AND then a 2? independent: P(A)·P(B)=1/2·1/6 = 1/12=0.083 Probability of 1 salmon getting through a dam is 0.85. What is the probability of 3 getting through the dam? Independent (.85)(.85)(.85)=0.614 Probability that none of the salmon get through? Probability that at least one gets through? Do TRY IT YOURSELF 4 on p. 118

20. Examples: Find the probabilities A coin is tossed AND a die is rolled. What is the probability of a head AND then a 2? independent: P(A)·P(B)=1/2·1/6 = 1/12=0.083 Probability of 1 salmon getting through a dam is 0.85. What is the probability of 3 getting through the dam? Independent (.85)(.85)(.85)=0.614 Probability that none of the salmon get through? failure = 1-.85 =.15 so P(none)=(.15)(.15)(.15)= 0.003 Probability that at least one gets through? Do TRY IT YOURSELF 4 on p. 118

21. Examples: Find the probabilities A coin is tossed AND a die is rolled. What is the probability of a head AND then a 2? independent: P(A)·P(B)=1/2·1/6 = 1/12=0.083 Probability of 1 salmon getting through a dam is 0.85. What is the probability of 3 getting through the dam? Independent (.85)(.85)(.85)=0.614 Probability that none of the salmon get through? failure = 1-.85 =.15 so P(none)=(.15)(.15)(.15)= 0.003 Probability that at least one gets through? Complement to None ( 1 or more) 1-P(none) = 1-.003 = 0.997 Do TRY IT YOURSELF 4 on p. 118

22. TRY IT YOURSELF 4 1. – A) event – B) P(3 successes)=(.9)(.9)(.9)=0.729 2. – A) complement – B) P(at least 1) = 1 – P(none) P(fail) = 1-.9 =.1 P(3 fail (none))= (.1)(.1)(.1)=.001 P(at least 1) = 1-.001 = 0.999

23. Assignment (Due Wed.) 3.2 p. 119 # 1-20