Question

Fifty-four percent of US teens have heard of a fax machine. You randomly select 12 US teens. Find the probability that the number of these selected teens that have heard of a fax machine is exactly six.

Option a: 0.217
Option b: 0.284
Option c: 0.120
Option d: 0.280

Answer 1

By applying the binomial probability formula, we find that the probability that exactly six out of twelve US teens have heard of a fax machine is approximately 0.285, which rounds to 0.280.

Step-by-step explanation:

To find the probability that exactly six out of twelve US teens have heard of a fax machine when 54% of US teens have heard of one, we should use the binomial probability formula:

P(X = k) = C(n, k) × p^k × (1-p)^(n-k)

Where:

C(n, k) is the number of combinations of n things taken k at a time (the binomial coefficient).

p is the probability of success on a single trial.

n is the number of trials.

k is the number of successes we're interested in.

In this problem, n = 12 (the number of teens), k = 6 (the number we're interested in who have heard of a fax machine), and p = 0.54 (the probability that a teen has heard of one).

Applying the formula, we get:

P(X = 6) = C(12, 6) × 0.54^6 × (1-0.54)^6

Now, let's calculate it:

C(12, 6) = 12! / (6! × (12 - 6)!)

Which is equal to:

924 combinations

Therefore, our final probability calculation will be:

P(X = 6) = 924 × (0.54)^6 × (0.46)^6 ≈ 0.285 (rounded to three decimal places)

The closest option to our calculated probability is Option d: 0.280, so that would be the best choice here.