2)Suppose you are invited to play a game. A ticket is picked from a box with 10

Question

2)Suppose you are invited to play a game. A ticket is picked from a box with 10 tickets numbered from one to ten. If you get a 2, 5, 7, you win three dollars. If you get anything else, you lose one dollar.

a)(2pts) Describe the box model for this game.

b)(4pts) How much do you expect to win/lose in 100 games? Find the standard error.

c)(3pts) What is the probability that you win at least $10?

d)(3pts) A fair game is one in which, on average, players neither win nor lose. Change the win-lose dollar amounts in the above game so that it is fair. There are many possible answers. You cannot change the winning numbers 2, 5, 7.

Explanation / Answer

a) In the box, 10 tickets number form 1 to 10.

If you get a 2, 5, 7, you win three dollars. If you get anything else, you lose one dollar.

b) Let X be amount when a game plays one time

X (amount) 3 1

P(X=x) 3/10 7/10

E(X) = 3*3/10 + 1*7/10 = 16/10 = 1.6

The expect to win in 100 games = 100*1.6 = $160

E(X2) = 32*3/10 + 12*7/10 =3.4

Var(X) = 3.4 - 1.62 = 0.84

Variance to win in 100 games = 100*0.84 = 84

SE = sqrt(84) = 9.16515

c)   The probability that you win at least $10 is P(X>=10) = P(Z> 10 - 160 /9.16515)

=P(Z>-16.3663) = 0

d) the game will be fair if E(X) = 0

Let X be amount when a game plays one time

X (amount) x -x

P(X=x) 3/10 7/10

Given E(X) = 0

x*3/10 -x *7/10 = 0

-4x = 0

x=0

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