# Probabilities Question

i have a Probabilities Question :

We are to find the probability that when three dice are rolled at the same time, the largest value of the three numbers rolled is 4.

let A be the outcome in which the largest number is 4, let B be the outcome in which the largest number is 4 or less, and let C be the outcome in which the largest number is 3 or less.

Let P(X) denote the probability that the outcome of an event is X .Then

(1) P(B) = ........ , * P(C) = ..........*

(2) since B = A U C and the outcomes A and C are mutually exclusive it follows that P(A) = .........

5 years, 1 month ago

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To calculate P(B), think about how many ways we can choose an ordered triplet (a,b,c) such that each of a, b, c is at most 4. How many choices can we make for a? How many choices can we make for b? How many choices can we make for c? A little thought should show you that there are 4 choices each, and they are all independent, so there should be $$4^3 = 64$$ such outcomes. Similarly, we can enumerate the outcomes corresponding to P(C); there should be $$3^3 = 27$$. Therefore, the number of desired outcomes that have a maximum of exactly 4 is equal to 64 - 27 = 37. How many unrestricted possible outcomes are there? $$6^3 = 216$$, so P(A) = 37/216.

This question is a good example of how calculating the probability mass function for an order statistic of a discrete probability distribution is made easier by considering the cumulative distribution function of the order statistic. That is to say, $$\Pr[X_{(k)} = x] = \Pr[X_{(k)} \le x] - \Pr[X_{(k)} \le x-1]$$ for a distribution whose support is a subset of the integers, and it is generally easier to compute the RHS rather than trying to enumerate the LHS directly.

- 5 years, 1 month ago