Thirty-two percent of adults did not visit their physicians' offices last year. The probability, rounded to
four decimal places, that in a random sample of 8 adults, exactly 3 will say they did not visit their
physicians' offices last year is:
P(X = 3) = nCx * p^x * (1-p)^n-x
8c3(.32)^3(1-.32)^8-3
8C3(.32)^3(.68)^5
=.2668
Well, aren't those adults just doctor-avoiding daredevils? Let's calculate the probability, shall we?
To solve this, we need to use the binomial probability formula. The formula is P(X=k) = nCk * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes, p is the probability of success, and nCk is the number of combinations.
So, in this case, n = 8 (since we have a sample size of 8 adults), k = 3 (since we want exactly 3 adults to say they didn't visit their physicians' offices), and p = 0.32 (the probability of an adult not visiting their physician's office).
Let me do the calculations for you:
P(X=3) = 8C3 * 0.32^3 * (1-0.32)^(8-3)
Let's find our calculators and do some math...
Calculating...
*drumroll please*
After some rigorous calculations, the probability, rounded to four decimal places, is approximately 0.1313.
So, the probability that exactly 3 out of 8 adults say they did not visit their physicians' offices last year is roughly 0.1313, or as some may say, about as likely as finding a clown riding a unicycle in a hospital waiting room.
To calculate the probability, we need to use the binomial probability formula. The formula for the probability of getting exactly k successes in n trials is:
P(x=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(x=k) is the probability of getting exactly k successes
- n is the total number of trials
- k is the number of successes
- p is the probability of success in a single trial
- (n C k) is the combination formula, which represents the number of ways to choose k successes from n trials
In this case, we have:
- n = 8 (random sample of 8 adults)
- k = 3 (exactly 3 will say they did not visit their physicians' offices last year)
- p = 0.32 (probability that an adult did not visit their physicians' office)
Let's plug in these values into the formula and calculate the probability:
P(x=3) = (8 C 3) * 0.32^3 * (1-0.32)^(8-3)
To calculate (8 C 3), we can use the combination formula:
(8 C 3) = 8! / (3! * (8-3)!)
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1
(8-3)! = 5 * 4 * 3 * 2 * 1
Now, let's calculate:
(8 C 3) = 8! / (3! * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
P(x=3) = 56 * 0.32^3 * (1-0.32)^(8-3)
Calculating this expression, we get:
P(x=3) = 0.3156 (rounded to four decimal places)
Therefore, the probability that in a random sample of 8 adults, exactly 3 will say they did not visit their physicians' offices last year is approximately 0.3156.
To find the probability, we need to use the binomial probability formula. The formula for the probability of getting exactly k successes in n independent Bernoulli trials, each with a probability of success p, is:
P(X = k) = (n choose k) * (p^k) * ((1 - p)^(n - k))
In this case, the probability of an adult not visiting their physician's office is given as 32% or 0.32. So, p = 0.32.
The number of successes we are interested in is exactly 3 out of 8 adults, so k = 3, and n = 8.
Now, we can substitute these values into the formula and calculate the probability:
P(X = 3) = (8 choose 3) * (0.32^3) * (0.68^5)
To simplify the calculation, we can use a calculator with binomial coefficient and exponent functions. The binomial coefficient can also be calculated as:
(8 choose 3) = (8!) / (3!(8-3)!), where ! denotes the factorial.
Calculating these values, we get:
P(X = 3) = (8 choose 3) * (0.32^3) * (0.68^5)
= (8!) / (3!(8-3)!) * (0.32^3) * (0.68^5)
= (8 * 7 * 6) / (3 * 2 * 1) * 0.032768 * 0.148035
= 0.273 * 0.032768 * 0.148035
≈ 0.0027
Therefore, the probability that exactly 3 out of 8 randomly selected adults will say they did not visit their physicians' offices last year is approximately 0.0027, rounded to four decimal places.