To solve this problem, we will use the binomial probability formula because we are calculating the probability of a specific number of "successes" (people who do not require vision correction) out of a fixed number of trials (7 people) with a constant probability of success for each trial.

Given Data

• Probability of an adult not requiring vision correction, $p = 0.24$
• Number of adults selected, $n = 7$
• Number of adults who do not require vision correction we want to find the probability for, $k = 2$

Binomial Probability Formula

The binomial probability formula is:

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}$

where:

• $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ is the binomial coefficient,
• $p$ is the probability of success on an individual trial,
• $k$ is the number of successes,
• $(1 - p)$ is the probability of failure on an individual trial.

Calculations

1. Binomial Coefficient: $\binom{7}{2} = \frac{7!}{2!(7 - 2)!} = \frac{7 \times 6}{2 \times 1} = 21$
2. Probability Component:
   • $p^k = 0.24^2 = 0.0576$
   • $(1 - p)^{n - k} = 0.76^5 = 0.2541$ (rounded to four decimal places)

Putting it all together:

$P(X = 2) = 21 \cdot 0.0576 \cdot 0.2541$

Calculating this:

$P(X = 2) \approx 21 \cdot 0.01464176 \approx 0.3075$

Final Answer

The probability that exactly 2 out of 7 randomly selected adults do not require vision correction is approximately 0.308 (rounded to three decimal places).

Would you like further details on any step, or do you have any questions?

Here are some related questions for further exploration:

1. How would the probability change if we wanted exactly 3 people not requiring vision correction?
2. What is the probability that no one out of the 7 adults requires vision correction?
3. How would this calculation change if 10 people were selected instead of 7?
4. How would you interpret this probability in a real-world scenario?
5. What is the expected number of people who do not require vision correction out of 7?

Tip: In binomial probability problems, checking for independent trials and fixed probabilities is crucial to ensure the formula applies correctly.