# 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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Chapter 5 / Lesson 5Businesses are faced with decisions every day, but how do you figure out the best course of action for your business? In this lesson, we'll explore two tools to help you out: decision trees and payoff tables.