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) = .........

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

hero p. - 8 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$