You are given 9 to 1 odds against tossing three heads with three coins, meaning you win $9 if you succeed and you lose $1 if you fail. (Find the expected value of the described game. Would you expect to win or lose money in 1 game? in 100 games? Explain.
You are given 55 to 44 odds against tossing three headsheads with three coins, meaning you win $55 if you succeed and you lose $44 if you fail. Find the expected value (to you) of the game. Would you expect to win or lose money in 1 game?
Well, to calculate the expected value, we need to find the probability of winning and losing.
The probability of tossing three heads with three coins is (1/2)^3 = 1/8, since there are 2 possible outcomes for each coin toss (heads or tails), and we want three heads in a row.
So, the probability of winning is 1/8, and the probability of losing is 1 - 1/8 = 7/8.
Now, let's calculate the expected value:
Expected value = (Probability of winning) x (Amount won) + (Probability of losing) x (Amount lost)
= (1/8) x ($9) + (7/8) x (-$1)
= $1.125 - $0.875
= $0.25
Therefore, the expected value of the described game is $0.25.
In one game, you would expect to win $0.25 on average. However, since this value is positive, you would expect to win money in the long run.
In 100 games, the expected value would be 100 x $0.25 = $25. So, you would still expect to win money over the course of 100 games.
Remember, though, that expected value is an average, and individual outcomes may vary. Good luck with your coin-tossing adventures!
To find the expected value, we multiply each possible outcome by its corresponding probability and sum them up. Let's break down the problem step-by-step.
Step 1: Determine the possible outcomes.
In this game, there are two possible outcomes:
- Succeed: Toss three heads with three coins.
- Fail: Not toss three heads with three coins.
Step 2: Determine the probabilities of each outcome.
To find the probability of each outcome, we need to consider the different ways in which three coins can land: head (H) or tail (T). Since each coin can have two possible outcomes (H or T), the total number of possible outcomes is 2 * 2 * 2 = 8. Out of these 8 possible outcomes, only 1 outcome will result in three heads (HHH). Therefore, the probability of success is 1/8. The probability of failure is 1 - 1/8 = 7/8.
Step 3: Calculate the expected value.
To calculate the expected value, multiply each outcome by its corresponding probability and sum them up:
Expected value = (Probability of success * Value of success) + (Probability of failure * Value of failure)
Expected value = (1/8 * $9) + (7/8 * (-$1))
Expected value = $9/8 + (-$7/8)
Expected value = $2/8
Expected value = $0.25
Step 4: Analyze the result.
The expected value of the game is $0.25. This means that, on average, you would expect to win $0.25 per game. In one game, you can't win or lose $0.25 since it is an average value. However, in the long run, over 100 games, you would expect to win $0.25 * 100 = $25. Therefore, overall, you should expect to win money in this game.
To calculate the expected value of a game, you need to multiply each outcome with its probability and sum them up. Let's break down the problem step by step:
1. Determine the probability of getting three heads with three coins:
There are two possible outcomes for each coin toss, heads (H) or tails (T). Since we have three coin tosses, there are a total of 2 * 2 * 2 = 8 possible outcomes, or 2^3.
Out of these 8 outcomes, there is only one outcome in which all three coins show heads: HHH. Therefore, the probability of getting three heads is 1/8.
2. Calculate the expected value:
The expected value (EV) is calculated by multiplying the outcome of each event with its probability and summing them up. In this case, the outcomes are winning $9 (W) or losing $1 (L), and their respective probabilities are 1/8 and 7/8 (since there are 7 out of 8 possible outcomes where you don't get three heads).
So, the expected value (EV) is:
EV = (W * P(W)) + (L * P(L))
= ($9 * 1/8) + (-$1 * 7/8)
= $9/8 - $7/8
= $2/8
= $0.25
3. Interpretation of the expected value:
The expected value of this game is $0.25. This means that, on average, you would expect to win $0.25 per game in the long run.
If you play the game once, you could either win or lose $1. However, over multiple games, the expected value suggests that you would, on average, win money. For example, in 100 games, you would expect to win approximately 100 * $0.25 = $25.
In summary, based on the expected value, you would expect to win money in the long run, although in any individual game, you might win or lose.
(1/8*9)-(7/8*1) = 1.125-.875 = .25
100 games you are expected to lose
1 game you are equally likely to win or lose