3 2 Conditional Probability The probability of an

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 • 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

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

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

TRY IT YOURSELF 1 • 1 a) # 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

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

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

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

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

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

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

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.

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

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

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

TRY IT YOURSELF 3 • 1. A = swim thru first B= swim thru 2 nd – A) independent – B) P(A and B) = P(A)·P(B) = (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

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

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

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)=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

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)=0. 614 • Probability that none of the salmon get through? failure = 1 -. 85 =. 15 so P(none)=(. 15)(. 15)= 0. 003 • Probability that at least one gets through? • Do TRY IT YOURSELF 4 on p. 118

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)=0. 614 • Probability that none of the salmon get through? failure = 1 -. 85 =. 15 so P(none)=(. 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

TRY IT YOURSELF 4 • 1. – A) event – B) P(3 successes)=(. 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)=. 001 P(at least 1) = 1 -. 001 = 0. 999

Assignment (Due Wed. ) • 3. 2 p. 119 # 1 -20