A certain insecticide kills $70 \%$ of all insects in laboratory experiments. A sample of 10 insects is exposed to the insecticide in a particular experiment. What is the probability that exactly 3 insects will die? Round your answer to four decimal places. Answer How to enter your answer (opens in new window) Tables Keypad Keyboard Shortcuts

Step 1 ：This problem can be solved using the binomial distribution formula. The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial, often referred to as 'success' and 'failure'. In this case, the 'success' is an insect dying.

Step 2 ：The formula for the probability mass function of a binomial distribution is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where \(P(X=k)\) is the probability of k successes, \(C(n, k)\) is the number of combinations of n items taken k at a time, p is the probability of success, n is the number of trials, and k is the number of successes.

Step 3 ：In this problem, the number of trials (n) is 10, the number of successes (k) we are interested in is 3, and the probability of success (p) is 0.7.

Step 4 ：First, we calculate the number of combinations of 10 items taken 3 at a time, which is 120.

Step 5 ：Next, we substitute these values into the formula to find the probability: \(P(X=3) = C(10, 3) * (0.7^3) * ((1-0.7)^(10-3))\).

Step 6 ：Calculating this gives a probability of 0.009.

Step 7 ：Final Answer: The probability that exactly 3 insects will die is \(\boxed{0.009}\).