A survey found 70% of adults over 50 wear glasses. In a random sample of 10 adults over 50, what is the probability that exactly six wear glasses and at least six wear glasses

To determine the probability, we need to use the concept of binomial probability. The binomial distribution is used when there are two possible outcomes (success or failure) for each trial, and the probability remains constant for each trial.

In this case, the success would be an adult over 50 wearing glasses, and the failure would be an adult over 50 not wearing glasses.

The probability of success (p) is 70% or 0.7, and the probability of failure (q) is 30% or 0.3.

To find the probability of exactly six adults wearing glasses out of a random sample of 10 adults over 50, we use the formula for binomial probability:

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

Where:
- P(X=k) represents the probability of getting exactly k successes
- (n C k) represents the combination of n items taken k at a time, which can be calculated as n! / (k! * (n-k)!)
- p^k represents the probability of k successes (in this case, exactly six adults wearing glasses)
- q^(n-k) represents the probability of (n-k) failures (in this case, four adults not wearing glasses)

We can substitute the values into the formula:

P(X=6) = (10 C 6) * (0.7^6) * (0.3^4)

Calculating this expression will give us the probability of exactly six adults out of the random sample wearing glasses.

To find the probability of at least six adults wearing glasses, we need to sum the probabilities of exactly six, seven, eight, nine, and ten adults wearing glasses:

P(X≥6) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

Calculating this expression will give us the probability of at least six adults out of the random sample wearing glasses.

To calculate the probability of exactly six adults wearing glasses in a random sample of 10 adults over 50, we can use the binomial probability formula:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

where:
P(X=k) represents the probability of exactly k successes
n is the number of trials
k is the number of successful events
p is the probability of a successful event (wearing glasses)
(1-p) is the probability of an unsuccessful event (not wearing glasses)
nCk represents the combination of n choose k (number of ways k successful events can occur in n trials)

In this case, n = 10, k = 6, and p = 0.7. Plugging these values into the formula, we have:

P(X=6) = (10C6) * 0.7^6 * (1-0.7)^(10-6)

Using the combination formula:
(10C6) = 10! / (6! * (10-6)!)

Simplifying:
(10C6) = 10! / (6! * 4!)
(10C6) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
(10C6) = 210

Now, we can calculate the probability:

P(X=6) = 210 * 0.7^6 * 0.3^4
P(X=6) ≈ 0.250822

Therefore, the probability of exactly six adults wearing glasses in a random sample of 10 adults over 50 is approximately 0.250822 or 25.08%.

To calculate the probability that at least six adults wear glasses in the same random sample of 10 adults over 50, we need to sum the probabilities for each possible outcome where k is greater than or equal to six.

P(X≥6) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

Using the binomial probability formula as mentioned earlier, we can calculate each individual probability and then add them together.

The probability of all events occurring is found by multiplying the probabilities of the individual events.

.7^6 * .3^4 = ? (exactly six)

Add probabilities of 6, 7, 8, 9 and 10 for "at least."

e.g., for 10 wearing = .7^10 = ?

for 9 wearing = .7^9 * .3 =?

I'll let you do the remaining calculations.