Question

Suppose that there are six corporations competing for seven different contracts. If the contracts are awarded randomly, what is the probability that each corporation will get at least one contract? (In exact rational fraction or decimal equivalent rounded to six decimal places)

Answer 1

Given, there are six corporations competing for 7 different contracts.

Let C₁ , C_{2 ,} C₃ , C4 , C_5, C₆ denote the 6 corporations.

we have to find probability that each corporation will get at least one contract.

There six different possibilities for this to happen.

One of the company will get 2 contracts and the other 5 will get on each.

i.e,

C₁ gets 2 contracts and the rest 5 gets 1 each OR

C2 gets 2 contracts and the rest 5 gets 1 each OR

C3 gets 2 contracts and the rest 5 gets 1 each OR

C4 gets 2 contracts and the rest 5 gets 1 each OR

C5 gets 2 contracts and the rest 5 gets 1 each OR

C6 gets 2 contracts and the rest 5 gets 1 each OR

Since the contracts are awarded randomly the required probability is given by

P(C₁ gets 2 contracts and the rest 5 gets one each) + P(C2 gets 2 contracts and the rest 5 gets one each) + P(C3 gets 2 contracts and the rest 5 gets one each) + P(C4 gets 2 contracts and the rest 5 gets one each) + P(C5 gets 2 contracts and the rest 5 gets one each) + P(C6 gets 2 contracts and the rest 5 gets one each)

= P(each of the corporations get 1 contract and any one of them get one more contract)

= 6*(6/7)*(1/7)

= 0.734694