Find the expected payback for a game in which you bet $9 on any number from 0 to 399. If your...

Question:

Find the expected payback for a game in which you bet $9 on any number from 0 to 399. If your number comes up, you get $500.

The expected payback is ${eq}\rule{0.5in}{0.3pt} {/eq}(round to the nearest cent as needed)

Expected Payoff

The expected payoff or payback is the weighted average of the payoffs from different results of a game. Each result has an associated probability and the weights are the associated probabilities. The expected payoff from a game is: $$\begin{align*} E(P) = \sum_{x_i \in X} p_ix_i \end{align*} $$ where {eq}X = \cup x_i {/eq} is the set of all possible outcomes of the game and {eq}p_i {/eq} is the associated probability of each event {eq}x_i {/eq}.

Answer and Explanation:

Given: A game of choosing any number from 0 to 399 for $9 and the prize for winning is $500.

The expected payoffs can be calculated by finding all possible outcomes of the game and their associated probabilities.

In this particular game, there are only two possible outcomes - a win or a loss. A win occurs when the chosen number matches your number, otherwise, it's a loss.

There are 400 numbers in the set 0..399 and the winning event occurs when exactly one number in this set comes up - which is the number you have.

So the probabilities of winning and losing are: $$\begin{align*} P(Win) &= \frac{1}{400} \\ \\ P(Loss) &= 1 - P(Win) \\ &= \frac{399}{400} \end{align*} $$

The payoff associated with winning is $$\begin{align*} X_{Win} &= $500 - $9,\quad\text{where \$9 is the cost of the ticket} \\ &= $441 \end{align*} $$.

And the payoff associated with losing is $$\begin{align*} X_{Loss} &= $0 - \$9 \\ &=-\$9 \end{align*} $$

So the expected payoff is $$\begin{align*} \text{Exp. Payoff} &= P(Win)X_{Win} + P(Loss)X_{Loss} \\ &= \frac{1}{400} \times \$441 - \frac{399}{400} \times $9 \\ &\approx -\$7.88 \end{align*} $$

The expected payoff is negative, which means that on average this is a losing bet.

Therefore:

The expected payback is {eq}\color{red}{-\$7.88} {/eq}.


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