The lottery commission has designed a new instant lottery game. Players pay $1.00 to scratch a ticket, where the prize won, X, (measured in $) has the following discrete probability distribution : $$\begin{array} { | c | c | }
\hline X & P [ X ] \
\hline 0 & 0.95 \
\hline 10 & 0.049 \
\hline 100 & 0.001 \
\hline
\end{array}$$ Which of the following best describes the expected value of X ?
A) In the long run, the average prize won per $\$ 1$ played is $\$ 0$.
B) In the long run, the average prize won per $\$ 1$ played is $\$ 0.41$.
C) In the long run, the average prize won per $\$ 1$ played is $\$ 0.59$.
D) None of these choices are correct

Question

The lottery commission has designed a new instant lottery game. Players pay $1.00 to scratch a ticket, where the prize won, X, (measured in $) has the following discrete probability distribution : $$\begin{array} { | c | c | } 
\hline X & P [ X ] \
\hline 0 & 0.95 \
\hline 10 & 0.049 \
\hline 100 & 0.001 \
\hline
\end{array}$$ Which of the following best describes the expected value of X ? 
A) In the long run, the average prize won per $\$ 1$ played is $\$ 0$.
B) In the long run, the average prize won per $\$ 1$ played is $\$ 0.41$.
C) In the long run, the average prize won per $\$ 1$ played is $\$ 0.59$.
D) None of these choices are correct

Quizplus · Accepted Answer

The expected value of X is given by the formula $E(X) = \sum_{i} x_{i} P[X=x_{i}]$, where $x_{i}$ and $P[X=x_{i}]$ are the possible outcomes and their respective probabilities. Using this formula, we have:

$E(X) = (0 	imes 0.95) + (10 	imes 0.049) + (100 	imes 0.001) = 0.49$

This means that in the long run, for every $1.00 played, the average prize won is $0.49. Therefore, the correct choice is C, which states that the average prize won per $1.00 played is $0.59, which is closest to $0.49.

The Lottery Commission Has Designed a New Instant Lottery Game XP[X]00.95100.0491000.001\begin{array} { | c | c | } \hline X & P [ X ] \\\hline 0 & 0.95 \\\hline 10 & 0.049 \\\hline 100 & 0.001 \\\hline\end{array}X010100​P[X]0.950.0490.001​​

The Lottery Commission Has Designed a New Instant Lottery Game $\begin{array} { | c | c | } \hline X & P [ X ] \\\hline 0 & 0.95 \\\hline 10 & 0.049 \\\hline 100 & 0.001 \\\hline\end{array}$