### Instruction: 8 coins are simultaneously flipped.  What is the probability that heads are showing on at most 2 of them?### Response: I need to find the probability of the event that at most 2 coins show heads, which means either 0, 1, or 2 coins show heads.
For each of these cases, I can use the binomial formula to calculate the probability: P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success in each trial.
In this problem, n = 8, p = 0.5 (since the coins are fair), and k can be 0, 1, or 2.
So I have:

P(X=0) = C(8,0) * 0.5^0 * (1-0.5)^(8-0) = 1 * 1 * 0.5^8 = 0.0039
P(X=1) = C(8,1) * 0.5^1 * (1-0.5)^(8-1) = 8 * 0.5 * 0.5^7 = 0.0313
P(X=2) = C(8,2) * 0.5^2 * (1-0.5)^(8-2) = 28 * 0.25 * 0.5^6 = 0.1094