Calculate the probability that at least 8 out of 10 animals will die from a drug, given that 4 out of 20 rats died from it.

This problem involves calculating probabilities for a fixed number of independent trials, where each trial has only two possible outcomes (death or survival). This is a binomial probability distribution. Here's the explanation and solution: 1. Identify the parameters of the binomial distribution: Number of trials (n): The number of animals treated with the drug, which is n = 10. Probability of success (p): The probability that an animal dies from the drug. Given that 4 out of 20 rats die, p = (4)/(20) = 0.2. Probability of failure (q): The probability that an animal survives. q = 1 - p = 1 - 0.2 = 0.8. Random variable (X): The number of animals that die. 2. State the binomial probability formula: The probability of exactly k successes in n trials is given by: P(X=k) = nk p^k q^n-k where nk = (n!)/(k!(n-k)!) is the binomial coefficient. 3. Determine the required probability: We need to find the probability that at least 8 animals will die. This means P(X 8), which is the sum of probabilities for X=8, X=9, and X=10. P(X 8) = P(X=8) + P(X=9) + P(X=10) Step 1: Calculate P(X=8). P(X=8) = 108 (0.2)^8 (0.8)^10-8 P(X=8) = (10!)/(8!2!) (0.2)^8 (0.8)^2 P(X=8) = (10 × 9)/(2 × 1) × (0.00000256) × (0.64) P(X=8) = 45 × 0.00000256 × 0.64 P(X=8) = 0.000073728 Step 2: Calculate P(X=9). P(X=9) = 109 (0.2)^9 (0.8)^10-9 P(X=9) = (10!)/(9!1!) (0.2)^9 (0.8)^1 P(X=9) = 10 × (0.000000512) × (0.8) P(X=9) = 0.000004096 Step 3: Calculate P(X=10). P(X=10) = 1010 (0.2)^10 (0.8)^10-10 P(X=10) = (10!)/(10!0!) (0.2)^10 (0.8)^0 P(X=10) = 1 × (0.0000001024) × 1 P(X=10) = 0.0000001024 Step 4: Sum the probabilities. P(X 8) = P(X=8) + P(X=9) + P(X=10) P(X 8) = 0.000073728 + 0.000004096 + 0.0000001024 P(X 8) = 0.0000779264 The probability that at least 8 animals will die is approximately 0.0000779. P(X 8) ≈ 0.0000779 3 done, 2 left today. You're making progress.