In a survey, 63% of Americans said they own an answering machine. If 14 Americans are selected at random, find the probability that exactly 9 own an answering machine.

Use a binomial probability function table. Values to look up are: p = .63, n = 14, and x = 9.

Or...

Calculate it by hand:
P(x) = (nCx)(p^x)[q^(n-x)]
...where q = 1 - p.

Well, let's do some probability calculations! We know that 63% of Americans own an answering machine, which means the probability of any one American owning one is 0.63.

Now, we want to find the probability that exactly 9 out of 14 Americans own an answering machine. This can be represented by a binomial distribution, where the probability of success (p) is 0.63, the number of trials (n) is 14, and we're interested in exactly 9 successes.

So, we can use the binomial probability formula:

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

Here, (n C k) represents the number of ways to choose k successes from n trials.

Plugging in the numbers, we have:

P(X=9) = (14 C 9) * (0.63^9) * (1-0.63)^(14-9)

Now, let me grab my calculator... *click*... *click*... *click*... *BEEP*

Calculating... calculating...

The probability of exactly 9 out of 14 Americans owning an answering machine is approximately 0.204.

So, the probability that exactly 9 out of 14 Americans own an answering machine is about 20.4%.

Now, that's a peculiar number of Americans to be interested in answering machines. Are they starting a band or something? Maybe they all got tired of answering each other's calls.

To find the probability of exactly 9 Americans owning an answering machine, we can use the binomial probability formula.

The binomial probability formula is given by:

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

Where:
P(X=k) is the probability of getting exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time,
p is the probability of success for a single trial, and
n is the total number of trials.

In this case, the probability of an American owning an answering machine is given as 63%, which can be expressed as 0.63. The total number of trials (Americans selected) is 14, and we want to find the probability of exactly 9 Americans owning an answering machine.

Plugging these values into the formula, we get:

P(X=9) = C(14, 9) * 0.63^9 * (1-0.63)^(14-9)

To calculate C(14, 9), we can use the combination formula C(n, k) = n! / (k!(n-k)!):

C(14, 9) = 14! / (9!(14-9)!)
= 14! / (9!5!)

Simplifying, we get:

C(14, 9) = (14 * 13 * 12 * 11 * 10) / (5 * 4 * 3 * 2 * 1)

Now we can calculate the probability:

P(X=9) = (14 * 13 * 12 * 11 * 10) / (5 * 4 * 3 * 2 * 1) * 0.63^9 * (1-0.63)^(14-9)

Finally, calculating this expression will give you the probability.

To find the probability of exactly 9 Americans out of 14 owning an answering machine, we can use the binomial probability formula. The formula is:

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

Where:
P(X = k) is the probability of k successes (9 owning an answering machine)
n is the number of trials (14 Americans selected)
k is the number of successes (9 owning an answering machine)
p is the probability of success in a single trial (63% = 0.63)
(1 - p) is the probability of failure in a single trial (1 - 0.63 = 0.37)

Now, let's calculate the probability:

P(X = 9) = (14 choose 9) * (0.63)^9 * (0.37)^(14-9)

(14 choose 9) = 14! / (9! * (14 - 9)!) = 2002

P(X = 9) = 2002 * (0.63)^9 * (0.37)^5

Using a calculator, you can evaluate this expression to find the exact probability.

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