Question

Suppose that there are four companies competing for five different government contracts. If the contracts are awarded randomly, what is the probability that each corporation will get at least one contract?

(a) 0.213

(b) 0.500

(c) 0.234

(d) 0.333

(e) 0.667

Answer 1

Solution: option (c) 0.234

Explanation :
The total ways (denominator) the contracts can be awarded is 5^4 because each of the 5 contracts
has 4 corporations that can be chosen. The number of ways they can be awarded so that each gets
at least one:
First, choose 4 of the 5 contracts to award to the 4 corporations, P(5,4). Then, award the last
contract to any one of the 4 corporations. So you have 4*P(5,4), but there are duplicates of each,
so divide by 2.
One challenge to this problem is that you are not asked to round, and the calculator does not
easily give you a simplified fraction. So, dividing by 4 as many times as possible, and then
dividing by 2, we have

4P(5,4)/2*4^5
P(5,4)/2*4^4
120/(2*256)
120/512
0.234