Question

Wayne is interested in two games, Keno and Bolita. To play Bolita, he buys a ticket for 1 marked with number 1..100, and one ball is drawn randomly from a collection marked with numbers 1,...,100. If his ticket number matches the number on the drawn ball he wins 75, otherwise he hey nothings and loses his 1 bet.

to play keno, he buts a ticket marked with numbers 1..4 and there are only 4 balls marked 1...4; again he wins if the ticket matches the ball draw. if he wins he gets 3, otherwise he loses his bet.

(a) What is the expected payout (expected value of net profit after buying ticket and possibly winning something) for each of these games?

(b) What is the variance and standard deviation for each of these games?

(c) If he decides to play one (and only one) of these games for a very long time, which one should he choose? If he decides to try one of these games for a couple of times, just for fun, which one should he choose?

Answer 1

Bolita game details:

Let x be the winning amount in each bet ,

hence if Wayne win the bet , his winning amount = Total win amount - bet amount = 75 -1 = 74

If Wayne lost the bet, his winning amount = 0 - 1 = -1

Out of 100 balls , only one ball is a winning ball, hence probability of winning a bet = 1/100 = 0.01

probability of loosing bet from compliment rule = 1 - probability of winning = 1 - 0.01 = 0.99

Bolita Game
Cost of each bet	Win amount (x)	P(X=x)	x * P(X=x)	* P(X=x)
1	74	1/100 = 0.01	0.74	54.76
1	-1	99/100 = 0.99	-0.99	0.99

Keno game details:

Let x be the winning amount in each bet ,

hence if Wayne win the bet , his winning amount = Total win amount - bet amount = 3 -1 = 2

If Wayne lost the bet, his winning amount = 0 - 1 = -1

Out of 4 balls , only one ball is a winning ball, hence probability of winning a bet = 1/4 = 0.25

probability of loosing bet from compliment rule = 1 - probability of winning = 1 - 1/4 = 3/4 = 0.75

Hence,

Keno Game
Cost of each bet	Win amount (x)	P(X=x)	x * P(X=x)	* P(X=x)
1	2	1/4 = 0.25	0.5	1
1	-1	3/4 = 0.75	-0.75	0.75

A)

from above table expected payout is evident from expected value in table , of both games that possibility of winning in each bet in both the games is ( -0.25).

B)

Bolita game :

from above table variance =

Keno game:

from above table variance , Keno game =

C)If he decides to play one (and only one) of these games for a very long time:

As Wayne is loosing same amount ( 0.25) in each bet for both games, the game which has less variance should be preferred. As Keno has 1.6875 variance , this game should be preferred by him for long time.

If he decides to try one of these games for a couple of times, he can prefer any one game, as both have same expected value (- 0.25) .