Question 151
A certain company wants to employ one secretary. Ten persons apply. The manager decides to interview them one by one according to the order of their applications. The first $3$ applicants should not be employed. From the fourth onward an applicant will be compared with the preceding ones. If he exceeds in ability all the preceding applicants, he will be employed. Otherwise he will not, and the interview goes on. If the preceding nine persons are not employed, the last one will be employed.
Suppose that the $10$ persons are different from each other in ability, and we can arrange them according to their ability rating from superior to inferior, such as $1$st, $2$nd, ..., $10$th. Obviously, whether an applicant will be eventually employed by the company depends on the order of the applications. As it is known, there are $10!$ such permutations in all. Now denote by $A_k$ the number of the permutations such that the applicant with the $k$-th ability rating is employed and the probability for him to be employed is $\frac{A_k}{10!}$. (posed by Su Chun)
Prove that under the policy given by the manager, we have the following properties:
$$(1) \quad A_1 \gt A_2 \gt \dots \gt A_8 = A_9 = A_{10}.$$
(2) The probability for the company to employ one of the persons with the ability among the top three is over 70%, and to employ one of the persons with the ability among the bottom three is not over 10%.
Suppose that the $10$ persons are different from each other in ability, and we can arrange them according to their ability rating from superior to inferior, such as $1$st, $2$nd, ..., $10$th. Obviously, whether an applicant will be eventually employed by the company depends on the order of the applications. As it is known, there are $10!$ such permutations in all. Now denote by $A_k$ the number of the permutations such that the applicant with the $k$-th ability rating is employed and the probability for him to be employed is $\frac{A_k}{10!}$. (posed by Su Chun)
Prove that under the policy given by the manager, we have the following properties:
$$(1) \quad A_1 \gt A_2 \gt \dots \gt A_8 = A_9 = A_{10}.$$
(2) The probability for the company to employ one of the persons with the ability among the top three is over 70%, and to employ one of the persons with the ability among the bottom three is not over 10%.