Question

Use the following statemnts to answer parts a) and b). Two hundred raffle tickets are sold for $3 each. One prize is $500 is to be awarded. Winners do not have their ticket costs $3 refunded to them. Raul purchases one ticket.

a) Determine his expected value

b) Determine the fair price of a ticket

(please and thank you so much I appreciate it)

Answer 1

So according to question we have to find E(x) .

Probability of winning $500 is 1/200 and not winning is 199/200 .

And E(X)=x_ip(x_{​​​​​​xi})

= 500 × (1/200) + (-3)×(199/200)

*[we have taken -3 , as it is mentioned that it is not refunded , so it can be regarded as a loss of $3 indicated by a negative sign ]*

= -0.485

So expected value is a loss of 0.485 $ .

Fair price = cost of playing the game + expected value earned from the game

Fair price of the ticket is = 3-0.485= 2.515 $

It can be done by other way too

Let the price be "x".
The random variable values are 500 and -x
The probabilities are 1/200 and 199/200
Multiplying and add you get:
500(1/200)+x(199/200) = 0
500 + 199x = 0
x = -(500/199) dollars
x = -2.5125 dollars
The price to play should be 2.515$ (approx)