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.

Step-by-step 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