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3.2-Conditional Probability

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. Examples.

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3.2-Conditional Probability

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  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

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