A certain brand of dinnerware set comes in three colors: red, white, and blue. Twenty percent of customers order the red set, 45% order the white, and 35% order the blue. Let X = 1 if a randomly chosen order is for a red set, let X = 0 otherwise; let Y = 1 if the order is for a white set, let Y = 0 otherwise; let Z = 1 if it is for either a red or white set, 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. Is it possible for both X and Y to equal 1?

e. Does pZ = pX + py ?

f. Does Z = X + Y ? Explain.

Answer:

Step 1 of 6:

(a)

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

Given brand of dinnerware has three colors: red, white, and blue.

Customers order the red set =

Customers order the white set =

Customers order the blue set =

Let if a randomly chosen order is for a red set, let otherwise.

Let if a randomly chosen order is for a white set, let otherwise.

Let if a randomly chosen order is for either a red or white set, let otherwise.

Since given data is in percentage.

We can say out of customers, customers ordered the red set.

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 .

We can say out of customers, customers ordered the red set.

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 a randomly chosen order is for either a red or white set, and otherwise.

Then we can write,

=

Substitute the values of probabilities from the above questions, and

=

=

Hence the success probability for () is .