answer:To find the probability of getting at least 4 heads in 8 flips, we can calculate the probability of getting exactly 4 heads, exactly 5 heads, exactly 6 heads, exactly 7 heads, and exactly 8 heads, and then add these probabilities together. However, a more elegant approach is to find the probability of the complementary event (getting 3 or fewer heads) and subtract it from 1. We already calculated the probability of getting exactly 3 heads in the previous question. To find the probability of getting at most 3 heads, we also need to consider the cases of getting 0, 1, or 2 heads. Using the binomial probability formula, we find that P(X = 0) = (8 choose 0) * (1/2)^0 * (1/2)^8 = 1/256, P(X = 1) = (8 choose 1) * (1/2)^1 * (1/2)^7 = 8/256, and P(X = 2) = (8 choose 2) * (1/2)^2 * (1/2)^6 = 28/256. We previously calculated P(X = 3) = 56/256. Adding these probabilities together, we get P(X ≤ 3) = (1 + 8 + 28 + 56)/256 = 93/256. Now, to find the probability of getting at least 4 heads, we subtract this value from 1: P(X ≥ 4) = 1 - P(X ≤ 3) = 1 - 93/256 = (256 - 93)/256 = 163/256. Therefore, the probability of getting at least 4 heads if you flip a fair coin 8 times is 163/256.

answer:The expected value of a random variable X, often denoted as E(X), represents the long-term average value that X is expected to take on. For a discrete random variable, we can calculate the expected value using the formula E(X) = ΣxP(X = x), where the sum is taken over all possible values of X. In this case, X can take on values from 0 to 8, corresponding to the number of heads in 8 flips of a fair coin. We know that the probability of getting exactly k heads is given by the binomial probability formula: P(X = k) = (8 choose k) * (1/2)^k * (1/2)^(8 - k). However, calculating the expected value by summing over all possible values of X and their corresponding probabilities would be tedious. Instead, we can use the fact that the expected value of a binomial random variable with n trials and probability of success p is given by E(X) = np. Here, we have n = 8 trials (the 8 flips of the coin) and p = 1/2 (the probability of getting a head). Plugging these values into the formula, we get E(X) = 8 * (1/2) = 4. This result makes intuitive sense: if we were to flip a fair coin 8 times many times, we would expect to get, on average, 4 heads each time. Therefore, the expected value of X is 4.

answer:<tool_call>[{name: normalize_filename, arguments: {filename: My Example File.txt}}]</tool_call>

answer:Michael shares success with modified basketball game and asks for resources on inclusive PE activities.