Questions: High blood pressure: A national survey reported that 34% of adults in a certain country have hypertension (high blood pressure). A sample of 24 adults is studied. Round the answers to four decimal places. (a) What is the probability that exactly 5 of them have hypertension?

Solution Steps

We are tasked with finding the probability that exactly 5 out of 24 adults have hypertension, given that the probability of an adult having hypertension is \( p = 0.34 \). The probability of not having hypertension is \( q = 1 - p = 0.66 \).

The probability of exactly \( x \) successes in \( n \) trials in a binomial distribution can be calculated using the formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Where:

• \( n = 24 \) (the number of trials),
• \( x = 5 \) (the number of successes),
• \( p = 0.34 \) (the probability of success),
• \( q = 0.66 \) (the probability of failure).

Using the values defined: \[ P(X = 5) = \binom{24}{5} \cdot (0.34)^5 \cdot (0.66)^{24-5} \]

Calculating \( \binom{24}{5} \): \[ \binom{24}{5} = \frac{24!}{5!(24-5)!} = \frac{24 \times 23 \times 22 \times 21 \times 20}{5 \times 4 \times 3 \times 2 \times 1} = 42504 \]

Now substituting back into the formula: \[ P(X = 5) = 42504 \cdot (0.34)^5 \cdot (0.66)^{19} \]

Calculating \( (0.34)^5 \) and \( (0.66)^{19} \): \[ (0.34)^5 \approx 0.0457 \quad \text{and} \quad (0.66)^{19} \approx 0.0001 \]

Thus: \[ P(X = 5) \approx 42504 \cdot 0.0457 \cdot 0.0001 \approx 0.072 \]

Final Answer