In this question, imagine a setting where two dice are about to be rolled. Assume the dice are standard, 6-sided dice, and are not weighted in way, so that each number is equally likely to be rolled. Consider the following 3 events:A is the event in which the sum of the numbers on the dice is ≤ 5.B is the event in which the sum of the numbers on the dice is ≥ 8.C is the event in which at least one of the dice is a 6.Question in Image Now, consider a random variable X defined as follows:6.if s E AX (s)2if s e Bn C-5otherwisef. Note that the three possible outcomes of the random variable are X = -5, X = 2, and X = 6. Calculatethe probability of each value of X.g. What is the expected value of the random variable, E(X)?

Answered: Now, consider a random variable X…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

In this question, imagine a setting where two dice are about to be rolled. Assume the dice are standard, 6-sided dice, and are not weighted in way, so that each number is equally likely to be rolled. Consider the following 3 events:

A is the event in which the sum of the numbers on the dice is ≤ 5.
B is the event in which the sum of the numbers on the dice is ≥ 8.
C is the event in which at least one of the dice is a 6.

Question in Image

Now, consider a random variable X defined as follows:
6.
if s E A
X (s)
2
if s e Bn C
-5
otherwise
f. Note that the three possible outcomes of the random variable are X = -5, X = 2, and X = 6. Calculate
the probability of each value of X.
g. What is the expected value of the random variable, E(X)?

Expert Solution

Step 1

According to the given information,

Let ' $S$ ' represent the sample space of rolling two dice together.

$S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}$

$n (S) = 36$

$A = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}$

$n (A) = 10$

$B = {(2, 6), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (5, 3), (5, 4), (5, 5), (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}$

$n (B) = 15$

$C = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}$

$n (C) = 11$

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Now, consider a random variable X defined as follows: 6. if s e A X (s) : if s e Bn C -5 otherwise f. Note that the three possible outcomes of the random variable are X = -5, X = 2, and X = 6. Calculate the probability of each value of X. g. What is the expected value of the random variable, E(X)?