Question

Five hundred chances are sold at 7 piece for a role. There is a grand price of $500, two second prizes of $275, and five third prizes of 100 First calculate the expected value of the bottery. Determine whether the lottery is a fair game. If the game is not fair, determine a price for playing the game that would make it The expected value of the game is (Type an integer or a decimal Round to the nearest cent) the game is not fair, determine a price for playing the game that would make fal 5 (Type an integer or a decimal Round to the nearest cent. Type F if the game is fale)

Answer 1

Solution-

According to question,

There are 500 chances that are sold at $ 7 each.

There is one grand price of $ 500 , two second prizes of $ 275 and 5 third prizes of $ 100 .

So,

one person win = $ (500 -7) = $493 each ...(1)

2 person win = $ (275 - 7) = $ 268 each ...(2)

5 person win = $ (100 - 7) = $ 93 each ...(3)

Probability of winning $ 493 = 1/500 ...(4)

Probability of winning $ 268 = 2/500 ...(5)

Probability of winning $ 93 = 5/500 ...(6)

So, there are 1 + 2 + 5 = 8 prizes in total.

It means out of 500 only 8 people win.

So, Number of people lose $ 7= 500 - 8 = 492

So, Probability of lossing $ 7 = 492/500 ...(7)

The Probability distribution is shown below-

Win/loss money (x) (in $)	Probability P(x)	xP(x)
493	1/500	493/500
268	2/500	536/500
93	5/500	465/500
-7	492/500	-3444/500
		$\sum xP(x)=-1950/500=- \$ \, 3.9$

Hene, expected value of the game is

$\sum xP(x)=- \$ \, 3.9$.

Since expected value of the game is negative. So, the game is unfair.

Game is fair if expected value is 0 dollars.

Let the prize of playing the game to make it fair is y dollars.

Now again.

Since,

There is one grand price of $ 500 , two second prizes of $ 275 and 5 third prizes of $ 100 .

So,

one person win = $ (500 -y) each ...(8)

2 person win = $ (275 - y) each...(9)

5 person win = $ (100 - y) each ...(10)

Probability of winning $ (500-y) = 1/500 ...(11)

Probability of winning $ (275-y) = 2/500 ...(12)

Probability of winning $ (100-y) = 5/500 ...(13)

Since, there are 1 + 2 + 5 = 8 prizes in total.

So, Number of people lose $ y is =500 - 8 = 492

So, Probability of lossing $ y = 492/500 ...(14)

The Probability distribution is shown below-

Win/loss (x) (in $)	P(x)	xP(x)
(500 - y)	1/500	(500 -y)/500
(275- y)	2/500	(550 - 2y)/500
(100 - y)	5/500	(500 -5y)/500
- y	492/500	-492y/500

Now, the expected value of [ xP(x) ] the game must be 0.

So,

$\sum xP(x)=0$

(500- y)/500 + (550 - 2y)/500 + (500 -5y)/500 - 492y/500 =0

Or 500/500 - y/500 + 550/500 - 2y/500 + 500/500 - 5y/500 - 492y/500 = 0

Or 500/500 + 550/500 + 500/500 -y/500 - 2y/500 - 5y/500 - 482y/500 =0

Or 1 + 1.1 + 1 - 500y/500 =0

Or 3.1- y = 0

Or y = 3.1 dollars.

Hence, for the game to be fair , the prize of playing the game must be $ 3.1 .