Question 1170267
Let's break down this problem step by step.

**Understanding the Problem**

This is a binomial probability problem because:

* There are a fixed number of trials (n = 16).
* Each trial has only two possible outcomes (high blood pressure or not).
* The probability of success (having high blood pressure) is constant (p = 0.20).
* The trials are independent.

**Given Information**

* Probability of having high blood pressure (success), p = 0.20
* Probability of not having high blood pressure (failure), q = 1 - p = 1 - 0.20 = 0.80
* Number of trials (sample size), n = 16

**Formula for Binomial Probability**

The probability of getting exactly k successes in n trials is:

P(X = k) = (nCk) * p^k * q^(n-k)

where:

* nCk = n! / (k! * (n-k)!) (the number of combinations of n items taken k at a time)

**a) None will have high blood pressure (k = 0)**

P(X = 0) = (16C0) * (0.20)^0 * (0.80)^(16-0)
P(X = 0) = 1 * 1 * (0.80)^16
P(X = 0) ≈ 0.0281

**b) Exactly four will have high blood pressure (k = 4)**

P(X = 4) = (16C4) * (0.20)^4 * (0.80)^(16-4)
P(X = 4) = (16! / (4! * 12!)) * (0.20)^4 * (0.80)^12
P(X = 4) = 1820 * 0.0016 * 0.068719476736
P(X = 4) ≈ 0.2001

**c) At least one will have high blood pressure (k ≥ 1)**

It's easier to find the probability of the complementary event (none have high blood pressure) and subtract it from 1:

P(X ≥ 1) = 1 - P(X = 0)
P(X ≥ 1) = 1 - 0.0281
P(X ≥ 1) ≈ 0.9719

**d) Mean and Standard Deviation**

For a binomial distribution:

* Mean (μ) = n * p
* Standard deviation (σ) = √(n * p * q)

Mean (μ) = 16 * 0.20 = 3.2

Standard deviation (σ) = √(16 * 0.20 * 0.80) = √(2.56) = 1.6

**Answers:**

a)  Approximately 0.0281
b)  Approximately 0.2001
c)  Approximately 0.9719
d)  Mean: 3.2, Standard deviation: 1.6