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

• 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?


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


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) = 50b) 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


• 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?





• Tossing a coin heads, then rolling a 6 on a 6 sided die?Independent : P(B|A)=1/6, P(B) = 1/6 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


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?


Examples:

• 2 cards are selected without replacement. What is the probability of a king AND then a queen? dependentP(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?


Examples:

• 2 cards are selected without replacement. What is the probability of a king AND then a queen? dependentP(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? independentP(H and 6) = P(H)·P(6)=1/2 · 1/6 = 1/12=.083


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

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


• 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?



• 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


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 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

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 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

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

Assignment (Due Wed.)

• 3.2 p. 119 # 1-20

3.2-Conditional Probability

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