Two dice are rolled. Let X = 1 if the dice come up doubles and let X = 0 otherwise. Let Y = 1 if the sum is 6, and let Y = 0 otherwise. Let Z = 1 if the dice come up both doubles and with a sum of 6 (that is, double 3), and let Z = 0 otherwise.

a. Let \(p_{X}\) denote the success probability for \(X\). Find \(p_{X}\) .

b. Let \(p_{Y}\) denote the success probability for \(Y\) . Find \(p_{Y}\).

c. Let \(p_{Z}\) denote the success probability for \(Z\). Find \(p_{Z}\).

d. Are \(X\) and \(Y\) independent?

e. Does \(p_{Z}=p_{X} p_{Y}\)?

f. Does Z = XY ? Explain.

Equation Transcription:

Text Transcription:

p_X

X

p_Y

Y

p_Z

Z

p_Z = p_X p_Y

Answer:

Step 1 of 6:

(a)

In this question, we are asked to find the success probability for and hence .

Two dice are rolled.

Let if the dice come up doubles and let otherwise.

Let if the sum is 6. and let otherwise.

Let if the dice come up both doubles and with a sum of 6, and let otherwise.

Let is the outcome of rolling the first dice where is the number from to that die will show.

Let is the outcome of rolling the second dice where is the number from to that die will show.

Dice are rolled in the pairs , we have of such pairs.

Since when the dice come up doubles, then the outcomes which contribute success are .

Hence number of outcomes where dice come up doubles =

Therefore the success probability is

=

Hence the success probability for () is .