A man comes up to play a game. you flip a coin, if heads you win $1 and keep playing if heads again you get $2, if you get heads again, $4 and so on. If you flip tails you take your money and the game is over. How much should he be willing to pay?
it depends onhow many times he flipes it
No it doesn't
It does matter how many time he flips it. E(x) changes with every trial added.
E(x)=Sum(P(u)X(u)) as u fulfills the total sample space
To determine how much the man should be willing to pay, we need to consider the expected value of the game. The expected value is the average amount of money the man can expect to win (or lose) per round of the game.
Let's analyze the game step by step:
1. If the man flips heads on the first toss, he wins $1.
2. If he flips heads again on the second toss, he wins an additional $2, for a total of $3.
3. If he flips heads a third time, he wins $4, making the total winnings $7.
4. This pattern continues for each subsequent toss.
However, if he flips tails at any point, he loses all the money he has accumulated thus far, and the game ends.
To calculate the expected value, we can consider the probability of each possible outcome. The probability of flipping heads on each toss is 1/2, as a fair coin has an equal chance of landing on heads or tails.
Let's calculate the expected value step by step:
1. The probability of flipping heads on the first toss is 1/2, with a corresponding winning amount of $1. Therefore, the expected value after the first toss is (1/2) * $1 = $0.50.
2. The probability of flipping heads on the second toss is also 1/2. Since it corresponds to a winning amount of $2, the expected value after the second toss is (1/2) * $2 = $1.
3. Continuing this pattern, the expected value after the third toss is (1/2) * $4 = $2.
Now, let's sum up the expected values:
Expected value after the first toss: $0.50
Expected value after the second toss: $1
Expected value after the third toss: $2
Since the pattern continues indefinitely, with the expected value doubling after each toss, we can express the expected value as follows:
Expected value = $0.50 + $1 + $2 + $4 + $8 + ...
This is an infinite geometric series with a first term of $0.50 and a common ratio of 2. The sum of an infinite geometric series can be calculated using the formula:
Sum = a / (1 - r)
where "a" is the first term and "r" is the common ratio.
In this case, plugging in the values:
Expected value = $0.50 / (1 - 2) = $0.50 / -1 = -$0.50
Therefore, the expected value of the game is -$0.50.
Since the expected value is negative, it means, on average, the man can expect to lose money per round of the game. Thus, he should not be willing to pay anything to play this game.