Question

A lottery ticket costs $2 and has a random 5-digit number. The payoff depends on the largest number of repeated digits. There is no prize if no digit is repeated. The prize is $2 if some digit is repeated exactly twice, $5 if a digit is repeated exactly 3 times, $10 if a digit is repeated exactly 4 times, and $1000 if a digit is repeated exactly 5 times. Let X be the net earning in playing the game.

a) Find the probability mass function of X.

b) Find the expectation E[X].

Answer 1

X	0		2	5	10	1000
outcomes	0		2	3	4	5
P(x)	0/14		2/14	3/14	4/14	5/14
	0		0.1428571	0.2142857	0.2857	0.357

b.

Mean

=sum(x * P(x))

= 0 * 0 + 2 * 0.14285 + 3*0.2142857 + 4 * 0.2857 + 5 * 0.357

= 3.856