point( $$ \$ $$ ) possible For a game of chance using a spinner, the probability of the event of winning when you pick a particular number is $$ \frac{1}{47} $$. Suppose you bet $$ \$ 1.00 $$ to play the game, and if your number is picked, you get back $$ \$ 44 $$ plus the $$ \$ 1 $$ that you bet. a. Is this a fair game? b. What would happen if you played this game a large number of times? a. Since the net winnings are b. You would expect to

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**Step 1. Define the outcomes and their probabilities**

- If you win, you receive $$ \$44 $$ additionally, but you also recover your original $$ \$1 $$ wager. The net gain is therefore $$ \$44 $$ (since you get $$ \$45 $$ in total but wager $$ \$1 $$ to play).
- If you lose, you lose your $$ \$1 $$ wager, so the net gain is $$ -\$1 $$.

The probability of winning is 
$$
\frac{1}{47}
$$
and the probability of losing is 
$$
\frac{46}{47}.
$$

**Step 2. Calculate the expected value (net winnings) per game**

The expected value $$ E $$ is computed as
$$
E = \left(\frac{1}{47}\right)(44) + \left(\frac{46}{47}\right)(-1).
$$

Simplify the expression:
$$
E = \frac{44}{47} - \frac{46}{47} = \frac{44 - 46}{47} = \frac{-2}{47}.
$$

This is approximately:
$$
\frac{-2}{47} \approx -0.04255 \text{ dollars per game.}
$$

**Step 3. Answer each part**

**(a) Is this a fair game?**

A game is fair if the expected value is $$ 0 $$. Here, since
$$
E \approx -0.04255,
$$
the game has a negative expected value for the player. Therefore, **this is not a fair game**.

**(b) What would happen if you played this game a large number of times?**

If you played the game many times, by the Law of Large Numbers, your average winnings per game would approach the expected value. This means **you would expect to lose about $$ \$0.04255 $$ per game on average**. Although the loss per game is small, over a large number of plays, these losses accumulate.
**(a) This game is not fair because the expected value is negative.** **(b) Over many plays, you would expect to lose approximately \$0.04255 per game on average.**

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