Based on the data collected by the National Centre for health Statistics and made available to the public in the Sample Adult database, it was found that the percentage of adults who have at some point in their life been told that they have hypertension was 15%. Suppose that a random sample of 12 adults is taken. What is the probability that exactly three have been told that they have hypertension?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
.15^3 * .85^9 = ?
To find the probability that exactly three out of twelve adults have been told that they have hypertension, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes,
n is the number of trials,
x is the number of successes,
p is the probability of success in a single trial, and
(1-p) is the probability of failure in a single trial.
In this case:
n = 12 (the number of adults in the sample),
x = 3 (the number of adults who have been told they have hypertension),
p = 0.15 (the probability of an adult having been told they have hypertension).
Using the formula, we can calculate the probability:
P(x = 3) = (12C3) * (0.15)^3 * (1-0.15)^(12-3)
To simplify this calculation, we can use the combination formula:
(12C3) = 12! / (3! * (12-3)!),
which equals 12! / (3! * 9!) = (12 * 11 * 10) / (3 * 2 * 1) = 220.
Plugging these values into the formula:
P(x = 3) = 220 * (0.15)^3 * (0.85)^(12-3)
P(x = 3) = 220 * (0.15)^3 * (0.85)^9
P(x = 3) ≈ 0.249
Therefore, the probability that exactly three out of twelve adults have been told they have hypertension is approximately 0.249 or 24.9%.
To find the probability that exactly three adults have been told that they have hypertension, we can use the binomial probability formula. The binomial probability formula is given by:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) represents the probability of exactly k successes,
- n represents the number of trials,
- p represents the probability of success in each trial, and
- (n choose k) is the binomial coefficient, given by n! / (k! * (n-k)!)
In this case, we know that n = 12 (the number of adults in the sample) and p = 0.15 (the probability of an adult having been told they have hypertension). We want to find P(X = 3), where X represents the number of adults in the sample who have been told that they have hypertension.
Using the formula, we can calculate the probability:
P(X = 3) = (12 choose 3) * 0.15^3 * (1-0.15)^(12-3)
Now, let's calculate these values step by step:
(12 choose 3) = 12! / (3! * (12-3)!) = (12 * 11 * 10) / (3 * 2 * 1) = 220
0.15^3 = 0.003375
(1-0.15)^(12-3) = 0.85^9 = 0.187387
P(X = 3) = 220 * 0.003375 * 0.187387 = 0.14049222
Therefore, the probability that exactly three adults in a random sample of 12 have been told that they have hypertension is approximately 0.1405 or 14.05%.