probability question

2007-02-28 7:55 am

Given that there are n distinct classes and k students (n > k), this k students will take m distinct classes (n > m) to attend. Assume they will choose the classes as random (i.e. for each students, the probability that he/she will choose one class is 1/n, no matter how many students have taken this class), what is the probability of there exist a class that no one choose, i.e.:
P(at least one class has no students) = ???

更新1:

Besides, what is the expected value of the number of students attending one class ?

回答 (1)

用戶113518

2007-02-28 6:00 pm

✔ 最佳答案

a) n classes, k students, and m classes for each student.

assume 1 SPECIFIC class that no one choose,
for the 1st student,
the 1st class chosen is (n-1) / (n)
the 2nd class chosen should be (n-2) / (n-1)
......
the m th class chosen should be (n-m) / (n-m+1)

So, the probability for the 1st student: (n-1)/(n) x (n-2)/(n-1) x (n-3)/(n-2) x ... x (n-m) / (n-m+1) = (n-m) / (n)
the probability for the 2nd student: (n-m) / (n)
......
the probability for the k th student: (n-m) / (n)

suming them up, the probability that 1 SPECIFIC class that no one choose:
(n-m)^k / (n)^k

there exist n classes, so the probability for ONE class that no one choose:
(n) (n-m)^k / (n)^k = (n-m)^k / (n)^(k-1)

b) the expected value of each class:
number of student: k
number of class chosen for each student: m
total number of class chosen: km
number of classes: n
so, the expected value for each class is km / n (evenly distributed)

參考: Myself

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