A recent survey found that 70% of all adults over 50 wear glasses for driving. In a random sample of 10 adults over 50, what is the probability that exactly eight wear glasses?

10/.70

0.850 To find the probability of exactly eight adults wearing glasses out of a random sample of 10 adults, we can use the binomial probability formula.

The binomial probability formula is given by:

P(x) = C(n, x) * p^x * (1-p)^(n-x)

Where:
P(x) is the probability of getting exactly x "successes"
C(n, x) is the number of possible combinations of n items taken x at a time (also called the binomial coefficient)
p is the probability of success (in this case, the probability of an adult wearing glasses)
n is the number of trials (in this case, the number of adults in the sample)
x is the number of successes (in this case, the number of adults who wear glasses)

In this case, p = 0.70 (70% probability of an adult wearing glasses) and x = 8 (exactly eight adults wear glasses).

To calculate the binomial coefficient C(n, x), we can use the formula:

C(n, x) = n! / (x! * (n-x)!)

Where ! denotes the factorial of a number.

Plugging in the values into the formula:

C(10, 8) = 10! / (8! * (10-8)!)
= (10 * 9 * 8!) / (8! * 2)
= (10 * 9) / 2
= 45

Now we can substitute the calculated values back into the binomial probability formula:

P(8) = 45 * (0.70)^8 * (1-0.70)^(10-8)
= 45 * (0.70)^8 * (0.30)^2
≈ 0.233

Therefore, the probability of exactly eight adults wearing glasses out of a random sample of 10 adults is approximately 0.233, or 23.3%.