Question

Eight hundred chances are sold at ​$7 apiece for a raffle. There is a grand prize of ​$500​, two second prizes of ​$200​, and five third prizes of ​$100. First calculate the expected value of the lottery. 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 fair. What is the expected value of the game?

Answer 1

solution:

Given data

800 chances are sold at $7 a piece for a raffle

There is a grand prize of ​$500​, two second prizes of ​$200​, and five third prizes of ​$100.

Therefore,The random gain values for 1st,2nd and 3rd prizes would be:

$493 , $193 , $93 and -$7

Let X be the randomvariable representing gain values,The distribution of X is

X	$493	$193	$93	-$7
P(X)	1/800	2/800	5/800	792/800

Expected value of lottery = E[X]

=

= (493 * 1/800 ) + (193 * 2/800) + (93 * 5/800) + (-7 * 792/800)

= [ 493+ 2*193 + 5*93 - 792*7 ] / 800

= -4200 / 500

= - $5.25

Therefore, Expected value of lottery = -$5.25

Here, The game is not fair as one can expected loose $5.25 in this game

The fair price of playing the game = $7 - $5.25 = $1.75 such that Expected value of this game equals to $0

X	$498.25	$198.25	$98.25	-$7
P(X)	1/800	2/800	5/800	792/800

Expected value of lottery = E[X]

=

= (498.25 * 1/800 ) + (198.25 * 2/800) + (98.25 * 5/800) + (-7 * 792/800)

= [ 498.25+ 2*198.25 + 5*98.25 - 792*7 ] / 800

= 0 / 500

= -$0

Therefore, Expected value of lottery = $0