Question

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability 0.3, and his second will lead independently to a sale with probability 0.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000 or the standard model, which costs $500. Determine the probability mass function of X, the total dollar value of all sales.

Answer 1

P(X=0) = 0.28

P(X=500) = 0.27

P(X=1000) = 0.315

P(X=1500) = 0.09

P(X=2000) = 0.045

Explanation:

Hi!

Lets think about the first apointment. There are three possible outcomes:

1. No sale (0 dollars)

2. Standard model (500 dollars)

3. Deluxe model (1000 dollars)

The probability of a sale is 0.3, and it could be Standard or Deluxe with equal probability. So, in the first apointment:

P(Standard) = 0.15

P(Deluxe) = 0.15

The same reasoning can be appiled to the second apointment. Lets define

a = dollars won in first apointment

b = dollars won in second apointment

Then we have:

P(a=0) = 0.7, P(a=500) = 0.15, P(a=1000) = 0.15

P(b=0) = 0.4, P(a=500) = 0.3, P(a=1000) = 0.3

The final outcome of the two apointments can be expressed as the pari (a,b) and X = a+b

As a and b are independent P(a,b) = P(a)*P(b)

What are the possible values of X? 0, 500, 1000, 1500, 2000

To find the probabilities of this values, we must find all the pairs that produce each value, and sum their probabilities (all the pairs are mutually exclusive). Then

P(X=0) = P(0,0) = 0.7*0.4 = 0.28

P(X=500) = P(500, 0) + P(0, 500) = 0.15*0.4 + 0.7*0.3 = 0.27

P(X=1000) = P(1000,0) + P(0,1000) + P(500,500) = 0.315

P(X=1500) = 0.09

P(X=2000) = 0.045

You can check that the sum of all these probabilities is 1.