Question

you are given 5 to 1 odds against tossing three tails with three coins, meaning you win \\$5 if you succeed and you lose \\$1 if you fail. find the expected value (to you) of the game. would you expect to win or lose money in 1 game? in 100 games? explain.

find the expected value (to you) for the game.
\\$\square$ (type an integer or a decimal rounded to the nearest hundredth as needed.)

would you expect to win or lose money in 1 game?
i would expect to win, because the probability of winning a game is greater than the probability of losing.
i would expect to lose, because the expected value is negative.
i would expect to win, because the expected value is positive.
d. i would expect neither because the outcome of one game cannot be predicted.

would you expect to win or lose money in 100 games?
o a. i would expect to lose, because the expected value is negative.
o b. i would expect to win, because the probability of winning a game is greater than the probability of losing.
o c. i would expect to lose, because the probability of losing a game is greater than the probability of winning.
o d. i would expect to win, because the expected value is positive.

Explanation:

Calculate the probability of winning and losing

Using the Odds Against knowledge point, odds against of \(5\) to \(1\) mean:

$$ P(\text{Loss}) = \frac{5}{5+1} = \frac{5}{6} $$

$$ P(\text{Win}) = \frac{1}{5+1} = \frac{1}{6} $$

Calculate the expected value of the game

Using the Expected Value knowledge point, we multiply each outcome value by its probability:

$$ E = (\$5) \cdot \frac{1}{6} + (-\$1) \cdot \frac{5}{6} = \frac{5}{6} - \frac{5}{6} = 0 $$

Thus, the expected value is \(\$0.00\).

Determine expectation in 1 game

Since the outcome of a single game is random, we cannot predict a win or loss.
Thus, we expect neither because the outcome of one game cannot be predicted.

Determine expectation in 100 games

Since the expected value is exactly \(0\), the long-term average net gain is \(0\).
Thus, we expect to break even (neither win nor lose money).

Answer:

Question 1

Find the expected value (to you) for the game.
<blank>\$0.00</blank>

Question 2

Would you expect to win or lose money in 1 game?

• D. I would expect neither because the outcome of one game cannot be predicted. (Correct answer)

Question 3

Would you expect to win or lose money in 100 games?

• C. I would expect neither because the expected value is 0. (Correct answer)