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Section 5.3-5.4

# 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)
= = ) 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
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. Exercise 1. Items coming off a production line are categorized as good (G), slightly blemished (B), and defective (D, and the percentages are 80%, 15% for good and for slightly blemished, respectively. Suppose that two items will be selected randomly for inspection and the selections are independent. (a)Find the probability that at least one of the items is slightly blemished. (b) Find the probability that neither of the items is good.

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

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