You are given 5 to 4 odds against tossing three heads with three coins, meaning you win $5 if you succeed and you lose $4 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.
P(HHH) = 1/8
So, E(x) = 1/8 * 5 - 7/8 * 4
To find the expected value, we need to calculate the probability of winning and the probability of losing.
The probability of getting three heads with three coins can be found using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (3 coins in this case)
- k is the number of successful outcomes (3 heads in this case)
- p is the probability of success in a single trial (0.5 for a fair coin)
Using the formula, we can calculate the probability of winning as:
P(Win) = P(X=3) = (3 choose 3) * 0.5^3 * (1-0.5)^(3-3) = 0.125
Therefore, the probability of losing is:
P(Lose) = 1 - P(Win) = 1 - 0.125 = 0.875
The expected value (EV) can be calculated as:
EV = (P(Win) * Win Amount) - (P(Lose) * Loss Amount)
Given the odds of 5 to 4, the Win Amount is $5 and the Loss Amount is -$4. Substituting the values:
EV = (0.125 * $5) - (0.875 * -$4)
= $0.625 + $3.5
= $4.125
Therefore, the expected value of the game is $4.125.
In 1 game, you would expect to win $4.125, so you can expect to win money.
In 100 games, if each game is independent, the expected value can be multiplied by the number of games:
EV (100 games) = $4.125 * 100
= $412.50
Therefore, in 100 games, you would expect to win approximately $412.50.
It's important to note that the expected value represents the average outcome over many trials. In any given game or a small number of games, the actual outcome may vary and may not match the expected value.
To find the expected value, we multiply the value of each possible outcome by its probability and sum them up. In this case, there are two possible outcomes: either you win $5 or you lose $4.
Let's calculate the probability of winning three heads with three coins:
- Each coin has two possible outcomes: either heads (H) or tails (T).
- With three coins, there are 2 * 2 * 2 = 8 possible outcomes.
- Out of these 8 outcomes, there is only 1 outcome where you get three heads (HHH).
So, the probability of tossing three heads with three coins is 1/8.
Now, we can calculate the expected value:
- The value of winning is $5, and the probability of winning is 1/8.
- The value of losing is -$4, and the probability of losing is 7/8 (since there are 7 outcomes where you don't get three heads).
Expected value = ($5 * 1/8) + (-$4 * 7/8)
Expected value = $5/8 - $28/8
Expected value = -$23/8
The expected value of the game is -$23/8, which means you can expect to lose approximately $2.88 in one game on average.
Now, let's consider 100 games. Since the expected value is negative, you would expect to lose money on average in each game. Therefore, in 100 games, you can expect to lose approximately 100 * $2.88 = $288.
In summary, based on the given odds and probabilities, you would expect to lose money in one game and also in 100 games, with an expected loss of $2.88 in one game and approximately $288 in 100 games.