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A penny and a nickel are tossed. Both are fair coins. Let

Problem 5E Chapter 4.1

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 5E

A penny and a nickel are tossed. Both are fair coins. Let X = 1 if the penny comes up heads, and let X = 0 otherwise. Let Y = 1 if the nickel comes up heads, and let Y = 0 otherwise. Let Z = 1 if both the penny and nickel come up heads, and let Z = 0 otherwise.

a. Let px denote the success probability for X. Find px.

b. Let pY denote the success probability for Y. Find pY.

c. Let pz denote the success probability for Z. Find pz.

d. Are X and Y independent?

e. Does pz=pxpy?

f. Does Z = XY? Explain.

Step-by-Step Solution:

Step 1 of 6:

(a)

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

Given penny and a nickel are tossed and both are fair coins.

Let if the penny comes heads, let   otherwise.

Let if the nickel comes heads, let   otherwise.

Let if both the penny and nickel come up heads, let   otherwise.

For tossing a coin, we can use the set {Heads, Tails} as the sample space

=

Hence the success probability for () is .

Step 2 of 6:

(b)

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

Since

=

Hence the success probability for () is .

Step 3 of 6:

(c)

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

Since if both the penny and nickel come up heads, and  otherwise.

Then we can write,

=

Substitute the values of probabilities from  the above questions,  and

=

=

Hence the success probability for () is .

Step 4 of 6

Step 5 of 6

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