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.

Answer:

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 .