Question

in a states lottery, you can bet $1 by selecting three digits, each between 0 and 9 inclusive. if the same three numbers are drawn in the same order, you win and collect $400. complete parts (a) through (e).
a. how many different selections are possible?
1000
b. what is the probability of winning?
$\frac{1}{1000}$
(simplify your answer.)
c. if you win, what is your net profit?
$399
(type an integer or a decimal. do not round.)
d. find the expected value for a $1 bet
$
(type an integer or a decimal. do not round.)

Explanation:

Step1: Define winning and losing cases

Let $x_1$ be the net - profit when winning and $x_2$ be the net - profit when losing. The probability of winning $p_1=\frac{1}{1000}$ and the probability of losing $p_2 = 1-\frac{1}{1000}=\frac{999}{1000}$. When winning, the net - profit $x_1=400 - 1=399$ (since the bet is $1$ and the payout is $400$), and when losing, the net - profit $x_2=- 1$.

Step2: Use the expected - value formula

The formula for the expected value $E(X)$ of a discrete random variable is $E(X)=\sum_{i = 1}^{n}p_ix_i$. Here, $n = 2$, so $E(X)=p_1x_1 + p_2x_2$. Substitute $p_1=\frac{1}{1000}$, $x_1 = 399$, $p_2=\frac{999}{1000}$, and $x_2=-1$ into the formula:
\[

$$\begin{align*} E(X)&=\frac{1}{1000}\times399+\frac{999}{1000}\times(-1)\\ &=\frac{399}{1000}-\frac{999}{1000}\\ &=\frac{399 - 999}{1000}\\ &=-\frac{600}{1000}=- 0.6 \end{align*}$$

\]

Answer:

$-0.6$