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).
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.
$-0.6
(type an integer or a decimal. do not round.)
e. if you bet $1 on a certain casino game, the expected value is - 3.5¢. which bet is better in the sense of producing a higher expected value: a $1 bet on the states lottery or a $1 bet on the casino game? explain.
a. neither bet is better because both games have the same expected value
b. the states lottery is a better bet because it has a larger expected value
c. it is impossible to compare the values because they have different units
d. the casino game is a better bet because it has a larger expected value

Explanation:

Step1: Recall expected - value concept

The expected value $E(X)$ of a discrete - random variable is calculated as $E(X)=\sum_{i}x_ip_i$, where $x_i$ are the possible values of the random variable and $p_i$ are their corresponding probabilities.

Step2: Analyze lottery situation

In the state's lottery, the probability of winning $p_w=\frac{1}{1000}$, and if you win, your net profit $x_w = 400 - 1=399$. The probability of losing $p_l=1 - \frac{1}{1000}=\frac{999}{1000}$, and if you lose, your net profit $x_l=-1$.

Step3: Calculate expected value of lottery bet

$E(X)=x_wp_w + x_lp_l=399\times\frac{1}{1000}+(-1)\times\frac{999}{1000}=\frac{399 - 999}{1000}=\frac{-600}{1000}=- 0.6$.

Step4: Compare expected values

The expected value of the state's lottery bet is $-0.6$ dollars and the expected value of the casino - game bet is $-0.035$ dollars. Since $-0.035>-0.6$, the casino game has a higher expected value.

Answer:

D. The casino game is a better bet because it has a larger expected value