Questions: [Question 7 of 10, Step 1 of 1 A certain insecticide kills 70% of all insects in a laboratory experiment. A sample of 71 insects is exposed to the insecticide that is randomly chosen. What is the probability that exactly 47 insects will survive? Round your answer to four decimal places. Answer How to enter your answer (opens in new window) Tables Keyboard Shortcuts]

Solution Steps

We are given a binomial experiment with parameters:

• Number of trials (n): 71
• Number of desired successes (k): 47
• Probability of success on a single trial (p): 0.3 We need to calculate the probability of exactly k successes.

The binomial probability formula is given by: $$ P(X = k) = \binom{n}{k} p^k q^{n-k} $$ where

• \(P(X = k)\) is the probability of getting exactly \(k\) successes in \(n\) trials,
• \(inom{n}{k}\) is the binomial coefficient, representing the number of ways to choose \(k\) successes out of \(n\) trials,
• \(p\) is the probability of success on a single trial,
• \(q\) is the probability of failure on a single trial (\(q = 1 - p\)).

Using the values provided, we calculate the probability as follows:

• The binomial coefficient \(inom7147\) is calculated as \( rac{n!}{k!(n-k)!} = 5.300174441392685\times 10^{18}\).
• Therefore, \(P(X = k)\) is \(5.300174441392685\times 10^{18} * (0.3^k) * (0.7^24) = 0\).

Final Answer: