### Instruction: If Anna flips 8 coins, what is the probability that she gets more heads than tails?### Response: To find the probability of an event, I need to divide the number of favorable outcomes by the total number of possible outcomes.
The total number of possible outcomes is 2^8, since each coin can land either heads or tails independently.
The number of favorable outcomes is harder to count, since I need to consider all the ways to get more heads than tails.
One way to approach this is to use combinations, which tell me how many ways to choose a subset of a given size from a larger set.
For example, if I want to get exactly 5 heads and 3 tails, I can think of choosing 5 coins out of 8 to be heads, and the rest to be tails.
The number of ways to do this is 8 choose 5, which is 8! / (5! 3!).
Similarly, if I want to get exactly 6 heads and 2 tails, I can think of choosing 6 coins out of 8 to be heads, and the rest to be tails.
The number of ways to do this is 8 choose 6, which is 8! / (6! 2!).
I can repeat this logic for 7 heads and 1 tail, and for 8 heads and 0 tails.
The number of ways to get 7 heads and 1 tail is 8 choose 7, which is 8! / (7! 1!).
The number of ways to get 8 heads and 0 tails is 8 choose 8, which is 8! / (8! 0!).
Now, to find the total number of favorable outcomes, I need to add up all these combinations.
This is 8 choose 5 plus 8 choose 6 plus 8 choose 7 plus 8 choose 8.
I can use a calculator or simplify the factorials to get the values of these combinations.
8 choose 5 is 56, 8 choose 6 is 28, 8 choose 7 is 8, and 8 choose 8 is 1.
So the total number of favorable outcomes is 56 + 28 + 8 + 1, which is 93.
Finally, I can divide this by the total number of possible outcomes, which is 2^8 or 256, to get the probability.
The probability is 93 / 256, which is about 0.363.