HW4-sol (Math 425) - = 2 = 5 10 × 5 9 P X = 3 = 5 10 × 4...

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Math 425 – Section 008 Homework 4 – Chapter 4, problems 1, 4, 18 Problem 4. Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman. Find P { X = i } , i = 1 , . . . , 10. Solution. Clearly, P { X > 6 } = 0. Now, suppose that X = i . This means that the first i - 1 ranked individuals are men, and the i th ranked is a woman. By using the multiplication rule, we get: P { X = 1 } = 5
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Unformatted text preview: = 2 } = 5 10 × 5 9 P { X = 3 } = 5 10 × 4 9 × 5 8 P { X = 4 } = 5 10 × 4 9 × 3 8 × 5 7 P { X = 5 } = 5 10 × 4 9 × 3 8 × 2 7 × 5 6 P { X = 6 } = 5 10 × 4 9 × 3 8 × 2 7 × 1 6 × 5 5 Alternative solution. Consider the event X ≥ i . It is equivalent to saying that the top i-1 individuals are men. Hence, P { X ≥ i } = ( 5 i-1 ) ( 10 i-1 ) Now, P { X = i } = P { X ≥ i } -P { X ≥ i + 1 } = ( 5 i-1 ) ( 10 i-1 )-( 5 i ) ( 10 i ) 1...
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