Question

two thousand raffle tickets are sold for $2 each. three prizes will be awarded, one for $1,000 and two for $500. assume that the probability that any given ticket is selected for the $1,000 prize is \\( \frac { 1 } { 2,000 } \\) and the probability that any given ticket is selected for a $500 prize is \LXI1. winners do not have their ticket costs of $2 refunded to them. jennifer purchases one of these tickets. complete parts (a) and (b) below. (a) determine jennifers expected value. $ - 1 (type an integer or a decimal rounded to the nearest hundredth as needed.) (b) determine the fair price of a ticket. $ (type an integer or a decimal rounded to the nearest hundredth as needed.)

Explanation:

Step1: Recall the formula for expected value

The formula for expected value \(E(X)=\sum_{i}x_ip_i\). Here, \(x_i\) are the possible values and \(p_i\) are their corresponding probabilities.
The possible values for Jennifer:

• If she wins the \(\$1000\) prize: Her net gain is \(x_1 = 1000 - 2=\$998\), and \(p_1=\frac{1}{2000}\)
• If she wins a \(\$500\) prize: Her net gain is \(x_2=500 - 2 = \$498\), and \(p_2=\frac{2}{2000}\)
• If she doesn't win: Her net gain is \(x_3=- 2\), and \(p_3 = 1-\frac{1 + 2}{2000}=\frac{1997}{2000}\)

Step2: Calculate the expected value for part (a)

$$
LATEXBLOCK0
$$

Step3: Recall the formula for fair - price

The fair price \(C\) of a ticket satisfies \(E(X)=0\). Let the fair price be \(C\).
The new possible values:

• If she wins the \(\$1000\) prize: Her net gain is \(x_1 = 1000 - C\), and \(p_1=\frac{1}{2000}\)
• If she wins a \(\$500\) prize: Her net gain is \(x_2=500 - C\), and \(p_2=\frac{2}{2000}\)
• If she doesn't win: Her net gain is \(x_3=-C\), and \(p_3=\frac{1997}{2000}\)

Since \(E(X) = 0\), we have \((1000 - C)\times\frac{1}{2000}+(500 - C)\times\frac{2}{2000}+(-C)\times\frac{1997}{2000}=0\)

Step4: Solve for \(C\) in part (b)

$$
LATEXBLOCK1
$$

Answer:

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