The probability that a randomly selected 1​-year-old male stink bug will live to be 2 years old is 0.95208.
​(a) What is the probability that two randomly selected 1​-year-old male stink bugs will live to be 2 years​ old?
​(b) What is the probability that eight randomly selected 1​-year-old male stink bugs will live to be 2 years​ old?
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Part 1
​(a) The probability that two randomly selected 1​-year-old male stink bugs will live to be 2 years old is

enter your response here.
​(Round to five decimal places as​ needed.)

To solve these problems, we'll use the basic principles of probability. If the probability of a single event occurring is \( P(A) \), then the probability of that event occurring \( n \) times independently is \( P(A)^n \).

Given:
- The probability that a randomly selected 1-year-old male stink bug will live to be 2 years old is \( 0.95208 \).

### Part (a)
To find the probability that two randomly selected 1-year-old male stink bugs will both live to be 2 years old, we calculate:

\[ P(\text{both survive}) = P(\text{one survives}) \times P(\text{other survives}) \]
\[ P(\text{both survive}) = 0.95208 \times 0.95208 \]

\[ P(\text{both survive}) = (0.95208)^2 \]

Using a calculator:

\[ P(\text{both survive}) = 0.95208^2 = 0.906456 \]

Rounded to five decimal places, the probability is:

\[ P(\text{both survive}) = 0.90646 \]

### Part (b)
To find the probability that eight randomly selected 1-year-old male stink bugs will all live to be 2 years old, we calculate:

\[ P(\text{all survive}) = (P(\text{one survives}))^8 \]
\[ P(\text{all survive}) = 0.95208^8 \]

Using a calculator:

\[ P(\text{all survive}) = 0.95208^8 \approx 0.698337 \]

Rounded to five decimal places, the probability is:

\[ P(\text{all survive}) = 0.69834 \]

### Summary
(a) The probability that two randomly selected 1-year-old male stink bugs will live to be 2 years old is \( 0.90646 \).

(b) The probability that eight randomly selected 1-year-old male stink bugs will live to be 2 years old is \( 0.69834 \).

Would you like more details or have any questions? Here are some other questions you might want to explore:

1. What is the probability that exactly one out of two stink bugs will live to be 2 years old?
2. How would the probabilities change if the survival rate was different?
3. What is the probability that at least one out of eight stink bugs will not live to be 2 years old?
4. How can we generalize the formula for any number of stink bugs?
5. What are the real-world applications of these probability calculations?

**Tip:** When dealing with probabilities of independent events, remember to multiply the probabilities of each individual event to find the combined probability.

Explore the probability calculations for stink bugs living to 2 years old. Calculate the probability for two and eight randomly selected 1-year-old male stink bugs based on a survival rate of 0.95208. Learn how to apply the formula for independent events in probability theory.

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