Probability Question 352
Question: If n objects are distributed at random among n persons, the probability that at least one of them will not get anything is
Options:
A) $ 1-\frac{(n-1)!}{{n^{n-1}}} $
B) $ \frac{(n-1)!}{n^{n}} $
C) $ 1-\frac{(n-1)!}{n^{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
The first object can be given to any of the n persons.
But the second, third and other objects, too, can go to any of the n persons.
Therefore the total number of ways of distributing the n objects randomly among n persons is $ n^{n} $ .
There are $ ^{n}P_{n}=n! $ ways in which each person gets exactly one object, so the probability of this happening is $ \frac{n!}{n^{n}}=\frac{(n-1)!}{{n^{n-1}}} $ .
Hence the probability that at least one person does not get any object is $ 1-\frac{(n-1)!}{{n^{n-1}}}. $
BETA
