Section 5.3-5.4 - 5.3 Conditional Probability and...

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5.3 Conditional Probability and Independence Conditional Probability Independent Events Parallel and Serial Systems Bayes’ Theorem Conditional Probability Let A, B two events. The conditional probability of B given condition A (given that A has occurred) is defined to be ) ( ) ( ) | ( A P B and A P A B P = or Example 1 A box contains 4 red and 2 green balls. Draw successively two balls without replacement and observe the color. Denote: G 1 = green on the first draw, G 2 = green on the second draw R 1 = red on the first draw, R 2 = red on the second draw 1 ) ( ) | ( ) ( ) ( ) | ( ) ( A P A B P B and A P B P B A P B and A P = =
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Sample space S = { G 1 G 2 , G 1 R 2, R 1 G 2 , R 1 R 2 } Probabilities P(G 1 G 2 ) = P(G 1 ) P(G 2 | G 1 ) = 2/6 ×1/5 = 2/30 P(R 1 R 2 ) = P(R 1 ) P(R 2 | R 1 ) = 4/6 ×3/5 =12/30 P(G 1 R 2 ) = P(G 1 ) P(R 2 | G 1 ) = 2/6 ×4/5 =8/30 P(R 1 G 2 ) = P(R 1 ) P(G 2 | R 1 ) = 4/6 ×2/5 =8/30 The probability that the second ball will be green. P(G 2 ) = P(R 1 G 2 ) + P(G 1 G 2 ) = 8/30+2/30 =10/30. The tree diagram is Suppose that green ball was observed in the second draw. Find the conditional probability that the first ball was also green. G 2 R 2 G 1 R 1 2 G1G2 G1 G1R2 R1G2 R1 R1R2 (1) (2)
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= = ) G ( ) G G ( ) G | G ( 2 2 1 2 1 P P P 2 . 0 ) 30 / 8 ( ) 30 / 2 ( 30 / 2 = + , since ). ( ) ( ) ( 2 1 2 1 2 G G P G R P G P + = Finally, compute the probability that exactly one ball selected is green, and probability that at least one ball selected is green (Do It Yourself!) Example 2. Given is a contingency table of 100 students cross- classified by their school goal and gender Goals Gender Grades Popular Sports Total Boy 24 10 13 47 Girl 27 19 7 53 Tota l 51 29 20 100 A student is selected at random. Let G ="a girl is selected" and S = "wants to excel at sports" 1. Find P(G) and Find P(S) . Note P(G)= (53/100)= 0.53 and P(S) = (20/100)= 0.2. 2. Find the probability that "a girl is selected and she wants to excel at sports" P(G and S) = (7/100) = 0.7. 3. Find the probability that "a student wants to excel at sports, given that a girl is selected ". 3
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Two events A and B are independent if Two events A and B are dependent if P(B|A) = P(B) or P(A and B) = P(A)P(B) P(B|A) P(B) or P(A and B) P(A)P(B) o o . 53 7 100 / 53 100 / 7 ) ( ) ( ) | ( = = = G P GS P G S P Independent events. The following table gives the definition Of independent events; Example 3. Let P(A) = 0.4 and P(B) = 0.5. If A and B are independent, find P(A and B) . Solution: Note P(A B)= P(A) P(B) = (0.4) (0.5)=0.2. 4
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Example 4 Given P(A) = 0.6, P(B) = 0.5, P(A or B) = 0.8. Are A and B independent? Note P(AB) = P(A) +P(B)- P( A B)= (0.6+ 0.5-0.8)= 0.3 = P(A) P(B). Hence, A and B are independent.
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This note was uploaded on 07/25/2008 for the course STT 351 taught by Professor Palaniappan during the Summer '08 term at Michigan State University.

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Section 5.3-5.4 - 5.3 Conditional Probability and...

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