Question

use the following statement to answer parts a) and b). one hundred raffle tickets are sold for $3 each. one prize of $200 is to be awarded. winners do not have their ticket costs of $3 refunded to them. raul purchases one ticket.
a) determine his expected value.
b) determine the fair price of a ticket.
a) his expected value is $ - 1 (type an integer or a decimal.)
b) the fair price of a ticket is $ (type an integer or a decimal.)

Explanation:

Step1: Calculate the probability of winning and losing

The probability of winning \(P(W)=\frac{1}{100}\), and the probability of losing \(P(L)=\frac{99}{100}\).
The amount won if winning is \(200 - 3=197\) (since the ticket cost \(3\) is not refunded), and the amount lost if losing is \(- 3\).

Step2: Calculate the expected value

The formula for expected value \(E(X)=\sum_{i}x_{i}P(x_{i})\).
\(E(X)=(197)\times\frac{1}{100}+(-3)\times\frac{99}{100}\)

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Step3: Calculate the fair price

Let the fair price of the ticket be \(x\). When the game is fair, the expected value is \(0\).
Let the amount won be \(200 - x\) (with probability \(\frac{1}{100}\)) and the amount lost be \(-x\) (with probability \(\frac{99}{100}\))
\(0=(200 - x)\times\frac{1}{100}+(-x)\times\frac{99}{100}\)

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Answer:

a) \(-1\)
b) \(2\)