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

Question

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)How much do you expect to win/lose in 100 games? Find the standard error. b)What is the probability that you win at least $10? c) 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) Expected value of one game = (probability of winning)*( 3 dollars) + (probability of loss)*( 1dollars)

= (3/10)*3 +(7/10)*(-1) =2/10 =.2

Since, each game can be considered indpendent, expected value of 100 games would be 100* Expected value of one game =100*2/10 =20

For calculating, standard error/deviation, lets calculcate variance first.

Variance= E(X^2)- (E(X))^2

Let's calculate E(X^2)

= (3^2)(3/10)+ (7/10)(-1)^2 =27/10 + 7/10 =34/10=3.4

Now, Variance= E(X^2)- (E(X))^2 = 3.4 -(.2)^2 =3.36

Hence, varince value of 100 games would be 100* variance value of one game =100*3.36 =336

Hence, standard deviation=sqrt(336) =18.33

(b) Assuming normal distribution, with standard deviation of 18.33 and mean value of 20, we will have to calculate the probability hat you win at least 10. Let's calculate z value..

Z=(10-20)/18.33 = (-.55)

Now, we have to calculate probability P( Z>= -.55) = .7088

(c) Let us assume that you will get x dollars on winning and y dollars on loosing.

Expected value of one game = (probability of winning)*( x dollars) + (probability of loss)*( -ydollars)

=(3/10)* x +(7/10)*(-y) =0 for a fair game

=> 3x=7y. So multiple answers are possible, if we assume x=7( winning) , y=3(loss).

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.