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, $p = \frac{4}{20} = 0.2$.
   • 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:

   $$P(X=k) = \binom{n}{k} p^k q^{n-k}$$

   where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

3. Determine the required probability: We need to find the probability that at least 8 animals will die. This means $P(X \ge 8)$, which is the sum of probabilities for $X=8$, $X=9$, and $X=10$.

   $$P(X \ge 8) = P(X=8) + P(X=9) + P(X=10)$$