7 men and 3 women are ranked according to their scores on an exam. Assume that no two scores are alike, and that all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a man (so X=1 indicates that a man achieved the highest score on the exam). Find each of the following:
P(X=1)=
P(X=2)=
P(X=3)=
P(X=7)=
To find the probabilities of each possible value of X, we need to calculate the number of favorable outcomes for each value and divide it by the total number of possible outcomes.
Let's start with P(X=1). In this case, a man achieves the highest score on the exam. Since there are 7 men and 3 women, we need to choose 1 man from the 7 available possibilities for the highest rank. The remaining 9 people can be arranged in any order. So, the number of favorable outcomes is 7 * 9!. The total number of outcomes is 10! (since there are 10 people in total). Therefore, P(X=1) = (7 * 9!) / 10!.
Next, let's move on to P(X=2). In this case, a man achieves the second-highest score on the exam. We need to choose 1 man from the 7 remaining possibilities for the second-highest rank. The highest rank can be given to any of the remaining 9 people. The remaining 8 people can be arranged in any order. So, the number of favorable outcomes is 7 * 9 * 8!. The total number of outcomes is still 10!. Therefore, P(X=2) = (7 * 9 * 8!) / 10!.
Similarly, we can calculate P(X=3) and P(X=7) using the same logic. For P(X=3), we choose 1 man from the 7 remaining possibilities for the third-highest rank, and then assign the remaining ranks to the remaining people. So, the number of favorable outcomes is 7 * 9 * 8 * 7!. The total number of outcomes is still 10!. Therefore, P(X=3) = (7 * 9 * 8 * 7!) / 10!.
Finally, for P(X=7), all the men achieve the lowest possible rank. We need to assign the top 3 ranks to the women and then assign the remaining ranks to the remaining people. So, the number of favorable outcomes is 3! * 7!. The total number of outcomes is still 10!. Therefore, P(X=7) = (3! * 7!) / 10!.
Now that we have the formulas for each probability, you can substitute the numbers and calculate them.