Suppose that 4 men and 4 women take an test and are then ranked from highest sco

Question

Suppose that 4 men and 4 women take an test and are then ranked from highest score (rank 1) to lowest score (rank 8). Suppose that no ties occur and that all orderings are equally likely. Let the random variable X = rank for highest-scoring woman.
a) List the possible values of X.
b) Determine the probability for each possible value of X. (As always, show/justify your calculations.)

are then ran from highest score (rank 1) to lowest score (rank 8). Suppose that no ties occur and that all orderings are equally likely. Let the random variable X- rank for highest-scoring woman. a) List the possible values of X. b) Determine the probability for each possible value of X. (As always, showljustify your calculations.)

Explanation / Answer

Solution:

(a) All possible values are X = 1 ,2,3,4,5,6,7,8 ( according to rank).

(b) We have 4 men and 4 women who take a test and are ranked according to their scores on an exam, where no two scores are alike and all 8! possible rankings are equally likely.  

Let X denote the highest ranking achieved by a woman. Clearly, since 5 is the lowest possible rank attainable by the highest-scoring female, we must have P(X = 6) = P(X = 7) =P(X = 8) = 0.

For P(X = 1)(female is a highest-ranking scorer), we have 4 possible choices out of 8 for the top spot that satisfy this requirement.

Hence, P(X = 1) = 4/8 = 1/2.

For P(X = 2)(female is 2nd-highest scorer), we have 4 possible choices for the top male, then 4 possible choices for the female who ranked 2nd overall, and then any arrangement of the remaining 6 individuals is acceptable (out of 8! possible arrangements of 8 individuals):

Hence,P(X = 2) = 4 *4 *6!/8! = 0.285.

For P(X = 3 )(female is 3rd-highest scorer), acceptable configurations yield (4)(3)= 12 possible choices for the top 2 males, 4 possible choices for the female who ranked 3rd overall, and 5! different arrangement of the remaining 5 individuals (out of a total of 8! possible arrangements of 8 individuals).

Hence, P(X = 3) = 12 * 5 *5!/8! = 0.178.

For P(X = 4) (female is 4th-highest scorer), acceptable configurations yield (4)(3)(2)=24 possible choices for the top 3 males, 4 choices for the female who ranked 4th overall, and 4! different arrangements of the remaining 4 individuals (out of a total of 8! possible arrangements of 8 individuals);

Here, P(X = 4) = 24*4 *4!/8! = 0.0571

For P(X = 5) (female is 5th-highest scorer), we basically have 4! arrangements of the males in the top 4 positions, and another 4! arrangements of the females in the bottom 4 positions; hence, the probability of all 4 males outperforming the top female is :

P(X = 5) = 4!*4!/8!=0.0142

Cheers!!

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