To calculate the expected winnings or losses, we need to find the probability of winning and losing. In this game, the probability of getting 6 tails or less in 13 coin flips can be calculated using binomial probability.
The probability of getting 6 tails or less can be found by summing the individual probabilities of getting 0, 1, 2, 3, 4, 5, or 6 tails.
Let's calculate it step by step:
1. Calculate the probability of getting exactly k tails:
- The probability of getting tails in a single coin flip is 1/2.
- The probability of getting heads in a single coin flip is also 1/2.
- The number of ways to arrange k tails in 13 coin flips is given by the binomial coefficient C(13, k).
- So, the probability of getting exactly k tails is (1/2)^k * (1/2)^(13-k) * C(13, k).
2. Sum up the probabilities of getting 0, 1, 2, 3, 4, 5, or 6 tails:
- Calculate the probability for each value of k and sum them up.
3. Multiply the probabilities by the corresponding winnings or losses:
- Multiply the probability of winning by $23 (positive value).
- Multiply the probability of losing by -$24 (negative value).
4. Calculate the expected winnings or losses:
- Sum up all the products from step 3.
Now, let's calculate the expected winnings or losses for playing this game 994 times.
