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?
To calculate the probability of exactly eight out of ten adults over 50 wearing glasses, we need to use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (n C x) * p^x * q^(n-x)
Where:
- P(x) is the probability of exactly x successes,
- n is the total number of trials,
- x is the number of successful trials,
- p is the probability of success in one trial,
- q is the probability of failure in one trial (q = 1 - p),
- (n C x) represents the number of combinations of n items taken x at a time.
In this case:
- n = 10 (the total number of adults over 50 in the sample),
- x = 8 (the number of adults who wear glasses),
- p = 0.70 (the probability of an adult over 50 wearing glasses),
- q = 1 - p = 0.30 (the probability of an adult over 50 not wearing glasses).
Now, we can substitute these values into the formula:
P(8) = (10 C 8) * (0.70)^8 * (0.30)^(10-8)
To calculate (10 C 8), we can use the binomial coefficient formula:
(10 C 8) = 10! / (8! * (10-8)!)
Simplifying further,
P(8) = (10! / (8! * 2!)) * (0.70)^8 * (0.30)^2
Calculating the factorials,
P(8) = (10 * 9 * 8!) / (8! * 2) * (0.70)^8 * (0.30)^2
Simplifying,
P(8) = 45 * (0.70)^8 * (0.30)^2
Now we can evaluate this expression to find the probability.