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 $ \frac { 2 } { 2,000 } $. winners do not have their ticket cost of $2 refunded to them. jennifer purchases one of these tickets. complete parts (a) and (b) below.
(a) determine jennifers expected value.
$\square$ (type an integer or a decimal rounded to the nearest hundredth as needed.)

Question

Accepted Answer

# Explanation:
## Step1: Calculate the net gain for each case
- **Winning the $1000$ prize**: The net gain is $1000 - 2=\$998$, and the probability $P_1=\frac{1}{2000}$.
- **Winning a $500$ prize**: The net gain is $500 - 2=\$498$, and the probability $P_2=\frac{2}{2000}$.
- **Not winning any prize**: The net gain is $- 2$ (since the ticket cost $2$ and no prize is won), and the probability $P_3 = 1-\frac{1}{2000}-\frac{2}{2000}=\frac{1997}{2000}$.

## Step2: Use the expected - value formula $E(X)=\sum_{i = 1}^{n}x_iP_i$
$$
\begin{align*}
E(X)&=(998)\times\frac{1}{2000}+(498)\times\frac{2}{2000}+(- 2)\times\frac{1997}{2000}\
&=\frac{998 + 498\times2-2\times1997}{2000}\
&=\frac{998 + 996-3994}{2000}\
&=\frac{1994 - 3994}{2000}\
&=\frac{-2000}{2000}\
&=- 1
\end{align*}
$$

# Answer:
$-1$

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Question

Explanation:

Step1: Calculate the net gain for each case

Step2: Use the expected - value formula \(E(X)=\sum_{i = 1}^{n}x_iP_i\)

Answer: