Question

A penny bank contains 1 quarter, 4 dimes, and 2 nickels. You reach in and randomly pull out two coins. Let X be the random variable representing the value of the coins you pull out in cents (e.g. a dime and a nickel would make X=15). Construct a probability distribution for X.

How many values of X are possible?

Answer 1

1 quarter=25 cent

1 dime =10 cents

1 nickel = 5 cents

Total number of coins in the box 1+4+2=7

Let N shows the event that a nickel is selected, D shows the event that a dime is selected and Q shows the event that a quarter is selected. Since order of coins is not important so possible outcomes are:

S = { NN, DD, NQ, ND, DQ}

The value of X in above cases is 10, 20, 30, 15, 35. Number of ways of seelcting 2 coins out of 7 is C(7,2) = 21. Since there are 2 nickels so

P(X = 10) = P(NN) = C(2,2) / 21 = 1/21

Since there are 4 Dimes so *

P(X = 20) = P(DD) = C(4,2) / 21 = 6/21

Likewise *

P(X = 30) = P(NQ) = [C(2,1)*C(1,1) ] / 21 = 2 / 21

P(X = 15) = P(ND) = [C(2,1)*C(4,1) ] / 21 = 8 / 21

P(X = 35) = P(DQ) = [C(1,1)*C(4,1) ] / 21 = 4 / 21

Following table shows the pdf:

X	P(X=x)
10	1/21
15	8/21
20	6/21
30	2/21
35	4/21