Q.4.4 Five men and 5 women are ranked ... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 4: Q.4.4 (page 163)

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. (For instance, $X = 1$ if the top-ranked person is female.) Find $P {X = i}, i = 1, 2, 3, 鈥�, 8, 9, 10$

Short Answer

Expert verified

$P (X) = {1 / 2, 5 / 18, 5 / 36, 5 / 84, 5 / 252, 1 / 252, 0, 0, 0, 0}$ for $x = 1, 2, 3 . . ., 10$ respectively.

Step by step solution

Step 1:Given Information

The total number of paths of ranking 10 different scores is 10!. The number of ways a female can be ranked 1 is.

ways of choosing any one out of 5. ways of arranging the rest 9

$P (X = 1) = \frac{^{5} C_{1} {鈰�}^{9} P_{9}}{^{10} P_{10}} = \frac{5.9!}{10!} = \frac{1}{2}$

Step 2:Explanation

The number of ways a female can be ranked 2 is,

Ways of female can be ranked 2 or less: ways that Ist male and female/5.Ways rest 8 are ranked.

P (X = 2) = \frac{^{5} P_{1} {鈰�}^{5} C_{1} {鈰�}^{8} P_{8}}{^{10} P_{10}} = \frac{5.5.8!}{10!} = \frac{5}{18}

Step 3:Explanation

$P (X = 3) = \frac{^{5} P_{2} {鈰�}^{5} C_{1} {鈰�}^{7} P_{7}}{^{10} P_{10}} = \frac{20.5.7!}{10!} = \frac{5}{36}$

Step 4:Explanation

$P (X = 4) = \frac{^{5} P_{3} {鈰�}^{5} C_{1} {鈰�}^{6} P_{6}}{^{10} P_{10}} = \frac{60 鈰� 5 鈰� 6!}{10!} = \frac{5}{84}$

Step 5:Explanation

$P (X = 5) = \frac{^{5} P_{4} {鈰�}^{5} C_{1} {鈰�}^{5} P_{5}}{^{10} P_{10}} = \frac{120.5.5!}{10!} = \frac{5}{252}$

Step 6:Explanation

$P (X = 6) = \frac{^{5} P_{5} {鈰�}^{5} C_{1} {鈰�}^{4} P_{4}}{^{10} P_{10}} = \frac{120 鈰� 5 鈰� 4!}{10!} = \frac{1}{252}$

Step 7:Explanation

There existing only $5$ boys, the lower value of $X$ can be $6$

P (X > 6) = 0

Step 8:Final Answer

$P (X) = {1 / 2, 5 / 18, 5 / 36, 5 / 84, 5 / 252, 1 / 252, 0, 0, 0, 0}$ for $x = 1, 2, 3, . . . 10$ respectively.

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91影视

Short Answer

Step by step solution

Step 1:Given Information

Step 2:Explanation

Step 3:Explanation

Step 4:Explanation

Step 5:Explanation

Step 6:Explanation

Step 7:Explanation

Step 8:Final Answer

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