Approximately 1 in 12 males of a certain ethnic descent has red-green color blindness. Suppose a group of 20 males of this descent is randomly chosen. Let random variable $X=$ the number of men in the group who have red-green color blindness. a) Find $P(X=3)$. What does this number represent? b) Find $P(X \geq 2)$. What does this number represent? c) Find $(3 \leq X \leq 6)$. What does this number represent? d) Find $\mu$ and $P(X=\mu)$. What do these numbers represent? a) What is the correct formula to use to find $P(X=3)$ ? \[ P(X=3)=\left(\frac{\square}{\square}\right) \cdot\left(\frac{1}{12}\right)^{\square} \cdot \square \square \text { (Simplify your answers.) } \] $P(X=3)=\square$, which is the probability that $\square$ males are color-blind. (Type an integer or decimal rounded to three decimal places as needed.) b) $P(X \geq 2)=\square$, which is the probability that $\square$ males are color-blind. (Type an integer or decimal rounded to three decimal places as needed.) c) $P(3 \leq X \leq 6)=\square$, which is the probability that males are color-blind. (Type an integer or decimal rounded to three decimal places as needed.) d) $\mu=\square$, where $\mu$ is the (Type an integer or decimal rounded to three decimal places as needed.) $P(X=\mu)=\square$, which is the probability that $\square$ males of the ethnic origin are color-blind. (Type an integer or decimal rounded to three decimal places as needed.) Next

Step 1 ：Define the parameters of the binomial distribution: the number of trials \(n = 20\) and the probability of success \(p = 1/12\).

Step 2 ：Calculate the probability \(P(X=3)\) using the binomial probability formula: \(P(X=3) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{n-k})\), where \(k = 3\). The result is approximately 0.150.

Step 3 ：This number represents the probability that exactly 3 males in the group are color-blind.

Step 4 ：Calculate the probability \(P(X \geq 2)\) by summing the binomial probabilities for \(k\) from 2 to \(n\). The result is approximately 0.505.

Step 5 ：This number represents the probability that at least 2 males in the group are color-blind.

Step 6 ：Calculate the probability \(P(3 \leq X \leq 6)\) by summing the binomial probabilities for \(k\) from 3 to 6. The result is approximately 0.229.

Step 7 ：This number represents the probability that between 3 and 6 males in the group are color-blind.

Step 8 ：Calculate the expected value \(\mu\) of the binomial distribution using the formula \(\mu = n \cdot p\). The result is approximately 1.667.

Step 9 ：\(\mu\) represents the expected number of males who are color-blind in the group.

Step 10 ：Calculate the probability \(P(X=\mu)\) using the binomial probability formula. Since \(\mu\) is not an integer, round it to the nearest integer before calculation. The result is approximately 0.319.

Step 11 ：This number represents the probability that approximately 2 males of the ethnic origin in the group are color-blind.

Step 12 ：Final Answer: a) \(P(X=3)=\boxed{0.150}\), which is the probability that 3 males are color-blind. b) \(P(X \geq 2)=\boxed{0.505}\), which is the probability that at least 2 males are color-blind. c) \(P(3 \leq X \leq 6)=\boxed{0.229}\), which is the probability that between 3 and 6 males are color-blind. d) \(\mu=\boxed{1.667}\), where \(\mu\) is the expected number of males who are color-blind. \(P(X=\mu)=\boxed{0.319}\), which is the probability that approximately 2 males of the ethnic origin are color-blind.