Suppose you toss a coin and will win $1 if it comes up heads. If it comes up tails, you toss again. This time you will receive $2 if it comes up heads. If it comes up tails, toss again. This time you will receive $4 if it comes up heads. Continue in this fashion for a total of 10 flips of the coin, after which you receive nothing if it comes up tails. What is the mathematical expectation for this game?
To calculate the mathematical expectation for this game, we need to determine the probability of each outcome and multiply it by the corresponding amount of money you would win.
In this game, you have a series of coin flips with increasing rewards each time you get heads. Let's break it down step by step:
1. On the first flip, you have a 50% chance of getting heads and winning $1. So the expected value for the first flip is (0.5 * $1) = $0.50.
2. If the first flip is tails, you proceed to the second flip. Here, you have a 50% chance of getting heads and winning $2. So the expected value for the second flip is (0.5 * $2) = $1.
3. Similarly, if the second flip is tails, you move on to the third flip. This time, you have a 50% chance of getting heads and winning $4. So the expected value for the third flip is (0.5 * $4) = $2.
4. You continue this process for ten flips, each time doubling the previous reward if it comes up heads. The expected value for each subsequent flip would then be: (0.5 * Previous Reward Doubled).
Now, to calculate the overall mathematical expectation, you sum up the expected values for each flip from the first to the tenth:
Expectation = $0.50 + $1 + $2 + $4 + $8 + $16 + $32 + $64 + $128 + $256
Expectation = $511.50
So, the mathematical expectation for this game is $511.50. This means, on average, you can expect to win $511.50 over the course of playing this game many times.