For a particular office telephone seven telephone calls are

For a particular office telephone, seven telephone calls are randomly received in a week, and these calls are randomly distributed among the seven days. What is the probability that no call will be received on exactly one day?

Solution

This is a very basic question of probability.

Let us first consider all the possible ways in which these 7 calls can occur in the week.

Assuming the office telephone works for all the 7 days and each call is distinct, each call has equal probability of coming on any day. So every call has 7 options of days to occur.

So, possible number of ways in which these calls can occur = 7⁷ = 823543.

Now, let us consider the case that no call will be received on exactly one day. That means other 6 days must get at least one call. So first of all let us select those 6 days. It will be - ⁷C₆ = 7 ways. That means the possible number of combinations of 6 days out of 7 days = 7. This can also be thought of the number of ways of leaving any one day of the week. So we can leave any of the Monday, Tuesday,...... or Sunday. So it can be done in 7 ways.

Now, the 7 distinct calls have to be distributed among these 6 days. So, any one day will have two calls and the other 5 days will have only 1 call. The possible number of ways in which it can occur can be calculated in two ways.

Method 1 - Let us first choose any 6 calls out of 7 and then find the number of permutations of these calls in 6 days. So, this will be - ⁷C₆ x 6! = 7! = 5040 ways.

Now, we have one call left, which can occur on any of these 6 days, so total possible number of ways will be -

5040 x ⁶C₁ = 30240.

Method 2 - we can also consider this case as grouping of 7 calls in 6 groups of 1,1,1,1,1 and 2 respectively with groups being distinct. So, the possible number of ways will be - (7! / (1!)⁵ x 5!) * 6! = 30240.

hence, the required probability = 30240 / 823543 = 0.0367

Hence, the probability that no call will be received on exactly one day = 0.0367