Question

you are given 6 to 1 odds against tossing three tails with three coins, meaning you win \\$6 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.
\\$ (type an integer or a decimal rounded to the nearest hundredth as needed.)

Explanation:

Calculate the probability of winning and losing

Using the Probability knowledge point
The event of interest is tossing three tails with three fair coins.
The sample space for tossing three coins has \(2^3 = 8\) equally likely outcomes:

$$ S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} $$

There is exactly \(1\) outcome with three tails (\(TTT\)).
Thus, the probability of winning (succeeding) is:

$$ P(\text{win}) = \frac{1}{8} = 0.125 $$

The probability of losing (failing) is:

$$ P(\text{lose}) = 1 - P(\text{win}) = \frac{7}{8} = 0.875 $$

Calculate the expected value of the game

Using the Expected Value knowledge point
The payout for winning is \(+\$6\) and the cost for losing is \(-\$1\).
The expected value \(E\) is calculated as:

$$ E = (\text{Payout for win} \times P(\text{win})) + (\text{Payout for lose} \times P(\text{lose})) $$

$$ E = (6 \times 0.125) + (-1 \times 0.875) $$

$$ E = 0.75 - 0.875 = -0.125 $$

Rounding to the nearest hundredth gives \(-\$0.13\).

Answer:

Find the expected value (to you) for the game.

<blank>\(-\$0.13\)</blank> (Type an integer or a decimal rounded to the nearest hundredth as needed.)