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Lecture 7(part II) - Lecture 7(part II Conditional...

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Lecture 7(part II) 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 = provided P(A)L0, or Example 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 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 ) ( ) | ( ) ( ) ( ) | ( ) ( A P A B P B and A P B P B A P B and A P = =
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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 Find 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 Tree diagram Suppose that green was observed in the second draw. Find the conditional probability that the first ball was also green. = = ) G ( ) G G ( ) G | G ( 2 2 1 2 1 P P 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!) G 2 R 2 G 1 R 1 G G R G R R (1) (2)
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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) L L Example: 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) 2. Find P(S) 3.
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