A certain insecticide kills 60%
 of all insects in laboratory experiments. A sample of 6
 insects is exposed to the insecticide in a particular experiment. What is the probability that exactly 2
 insects will survive? Round your answer to four decimal places.

This problem involves a binomial distribution because there are two outcomes for each insect: either it survives or it doesn't. The probability of exactly 2 insects surviving can be calculated using the binomial probability formula:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}
\]

Where:
- \( n = 6 \) (the number of insects),
- \( k = 2 \) (the number of surviving insects),
- \( p = 0.4 \) (the probability of survival, since 60% die, 40% survive),
- \( 1 - p = 0.6 \) (the probability of death).

We can now plug in the values:

\[
P(X = 2) = \binom{6}{2} (0.4)^2 (0.6)^4
\]

First, let's calculate \(\binom{6}{2}\) (the number of ways to choose 2 survivors from 6 insects):

\[
\binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \times 5}{2 \times 1} = 15
\]

Now substitute back into the formula:

\[
P(X = 2) = 15 \times (0.4)^2 \times (0.6)^4
\]

Let's compute the result.The probability that exactly 2 insects will survive is \( 0.3110 \), rounded to four decimal places.

Would you like more details on this calculation, or do you have any other questions?

Here are some related questions:
1. What is the probability that all 6 insects will survive?
2. What is the probability that none of the insects will survive?
3. What is the expected number of surviving insects in this experiment?
4. What is the probability that at least 4 insects will survive?
5. How does the binomial distribution apply to different sample sizes?

**Tip:** The expected value for the number of successes in a binomial experiment is \( E(X) = np \).

A certain insecticide kills 60% of all insects in laboratory experiments. A sample of 6 insects is exposed to the insecticide in a particular experiment. What is the probability that exactly 2 insects will survive? Round your answer to four decimal places.

Discover how to calculate the probability that exactly 2 insects will survive when exposed to an insecticide that kills 60% of them. Using the binomial probability formula and key concepts of probability and binomial distribution, this problem is perfect for students in Grades 10-12.

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