Let me analyze the content of the image and provide solutions to the mathematical problems presented. Please hold on for a moment.Here is the analysis and solutions for the questions visible in the image:

Question 7

Which of the following random variables is geometric?

• A geometric random variable counts the number of trials needed for the first success in a sequence of independent trials, each with the same probability of success.

Answer: (C) The number of times I have to roll a six-sided die to get two 5’s is not geometric, as it involves more than one success. However, (E), the number of trials needed to get the first 5, is a geometric random variable.

Question 8

Suppose 30% of the employees in a large factory are smokers, and a 5-person workforce is randomly chosen. What is the probability that less than three employees will be smokers?

This follows a binomial distribution, where $n = 5$, $p = 0.3$, and we seek $P(X < 3)$, meaning $P(X = 0) + P(X = 1) + P(X = 2)$.

1. $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
2. Compute:
   • $P(X = 0) = \binom{5}{0}(0.3)^0(0.7)^5 = 0.16807$
   • $P(X = 1) = \binom{5}{1}(0.3)^1(0.7)^4 = 0.36015$
   • $P(X = 2) = \binom{5}{2}(0.3)^2(0.7)^3 = 0.3087$

Summing these: $P(X < 3) = 0.16807 + 0.36015 + 0.3087 = 0.837$

Answer: (C) 0.837

Question 9

When is the sampling distribution of $X$ approximately normal?

The sample size $n = 100$ is large, and the proportion $p$ (success probability) needs to satisfy: $n \cdot p \geq 10 \quad \text{and} \quad n \cdot (1 - p) \geq 10$ Here:

• $n \cdot p = 100p$, and $n \cdot (1-p) = 100(1-p)$.

For the sampling distribution to be approximately normal, $p$ must lie within $0.1 \leq p \leq 0.9$.

Answer: (D) $p = 0.08$ is too low, so all other options are valid.

Question 10

Supermarket scanner accuracy:

The scanner correctly prices 99.8% of items. The number of scans until the first error follows a geometric distribution with $p = 0.002$.

The expected value for a geometric distribution is: $E(X) = \frac{1}{p} = \frac{1}{0.002} = 500$

Answer: (C) 500

Question 11

The probability that more than 4 contaminated chickens exist in a crate (24 chickens, 30% contaminated):

This follows a binomial distribution with:

• $n = 24$, $p = 0.3$, $P(X > 4)$.

Using the complement rule: $P(X > 4) = 1 - P(X \leq 4)$

Using a binomial calculator: $P(X \leq 4) = \sum_{k=0}^4 \binom{24}{k} (0.3)^k (0.7)^{24-k} = 0.0424$

Then: $P(X > 4) = 1 - 0.0424 = 0.9576$

Answer: (E) 0.9576

Question 12

Mean and standard deviation of contaminated chickens (24 chickens, $p = 0.3$):

For a binomial distribution: $\mu = n \cdot p = 24 \cdot 0.3 = 7.2$ $\sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{24 \cdot 0.3 \cdot 0.7} = \sqrt{5.04} \approx 2.24$

Answer: (D) $\mu = 7.2, \sigma = 2.24$

Would you like further clarification on any question? Here are 5 related questions to deepen understanding:

1. What is the difference between a geometric and binomial random variable?
2. How can the complement rule simplify solving probability questions?
3. Why is the sample size important in determining normality for sampling distributions?
4. Can you explain the derivation of mean and variance for a binomial distribution?
5. How would the answers change if the contamination rate for chickens were 50% instead of 30%?

Tip: Always identify whether a problem involves binomial, geometric, or normal distributions early to simplify the solution process.