Question

A certain brand of trading cards is available in smaller packs that cost $2 each. Assume that any given pack has a 10% chance of containing a rare card, independently of other packs. Alice will buy and open a pack until she gets a rare card, but she will not buy more than five. Let M be the amount of money (in dollars) that Alice will spend. a. Find the distribution of the random variable (and provide it as a chart).b. Calculate the expected value of the random variable. Do not round values.

Answer 1

Explanation:

• Let M = the amount of money alice will spend

• m takes values ; 2, 4, 6

• from what was given, P(m =2) = 0.10

• P(m=4) = the second rack contain rare card given that the first does not contain

• = (1 - 0.10) x 0.10 = 0.09

P(m=6) = 0.10 x 0.90^2 = 0.081

P(m=8) = 0.10 x 0.90^3 = 0.0729

P(m=10) = 1- [p(m=2) + p(m=4) + p(m=6) + p(m=8)

= 1 - 0.3439

= 0.6561

b) the expected value of the random variable ; E(M) = Summation(Px)

= 2x0.10 + 4x0.09 + 6x0.081 + 8x0.0729 + 10x0.6561

= 8.1902

The distribution of the random variable =

M P(M)

2 0.10

4 0.09

6 0.081

8 0.0729

10 0.6561