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Lecture%2011

# Lecture%2011 - Section 4.4 Multiplication rule In this...

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Section 4.4 Multiplication rule In this section, the event A and B will mean: ( A occurs in the first trial ) and ( B occurs in the second trial ) . Suppose there are two questions in an exam: 1. (true or false) Abraham Lincoln was the president during Civil War. 2. (check the correct answer) 5 + 7 = (a) 9 (b) 11 (c) 12 (d) 16 (e) 22 Suppose someone guesses both answers. What is the probability that both are correct? Let us do it hard way: Write the sample space: (T,a), (T,b), (T,c), (T,d), (T,e), (F,a), (F,b), (F,c), (F,d), (F,e) There is only one way both answers are correct, i.e., (T,c) P( both answers are correct ) = P( T and c ) = 1 / 10 = 0.1

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Suppose there are two questions in an exam: 1. (true or false) Abraham Lincoln was the president during Civil War. 2. (check the correct answer) 5 + 7 = (a) 9 (b) 11 (c) 12 (d) 16 (e) 22 Note: P(T) = 1 / 2 P(c) = 1 / 5 P(T) * P(c) = (1 / 2) * (1 / 5) = 1 /10 = P( T and c ) Based on above, can we say P( A and B ) = P(A) * P(B) ??? Yes and No !!! We have to look at it carefully .
Polygraph test result: Person did not lie Person lied Total Positive test result 15 42 57 (test indicated the person lied) (false positive) (true positive) Negative test result 32 9 41 (test indicated the person did not lie) (true negative) (false negative) Total 47 51 98 Problem: 2 subjects are chosen at random, without replacement . Find the probability that 1 st person had a positive result and the 2 nd person had a negative result First selection: P( positive result ) = 57 / 98 Second selection: P( negative result) = 41 / 97 We are using without replacement . After the 1 st selection of the subject with positive result, there are 97 subjects remaining and 41 of the remaining had negative result. Now, P( 1 st subject had positive result and 2 nd subject had negative result) = (57/98) * (41/97) = 0.246

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Important to note: We must adjust the probability of the second event to reflect the outcome of the first event If the 1 st event is A and the second event is B, the probability of the 2 nd event B should take into account that the 1 st event A has already occurred. Conditional probability: P ( B | A ) means probability of the event B, where it is known that the event A has already occurred. B | A means ``B given A’’ i.e., event B occurs a fter the A has already occurred. Multiplication rule: P( A and B ) P(A) * P(B) in general (it can happen in special cases) The real multiplication rule is: P( A and B ) = P(A) * P( B | A)
Polygraph test result: Person did not lie Person lied Total Positive test result 15 42 57 (test indicated the person lied) (false positive) (true positive) Negative test result 32 9 41 (test indicated the person did not lie) (true negative) (false negative) Total 47 51 98 We are using without replacement : P( negative result) = 41 / 98 But …. P( 2

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Lecture%2011 - Section 4.4 Multiplication rule In this...

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