the percentage of adults who have at some point in their life been told that they have hypertension is 23.53%. If 8 adults are randomly selected, find the probability that between 2 and 4 of then inclusive have been told that they have hypertension
To find the probability that between 2 and 4 of the 8 randomly selected adults have been told they have hypertension, we first need to find the probability for each specific case and then sum them up.
Let's calculate them step by step:
Step 1: Calculate the probability that exactly 2 adults have been told they have hypertension.
The probability for each specific case is calculated using the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where
- n is the total number of trials (8 in this case),
- k is the number of successful trials (2 in this case),
- C(n, k) is the number of combinations of n items taken k at a time (8 choose 2), and
- p is the probability of success for each trial (23.53%).
Using these values in the formula, we can calculate the probability for exactly 2 adults:
P(X=2) = C(8, 2) * (0.2353)^2 * (1-0.2353)^(8-2) = 28 * 0.0553 * 0.7658 ≈ 0.1214
Step 2: Calculate the probability that exactly 3 adults have been told they have hypertension.
Using the same binomial probability formula, we can calculate the probability for exactly 3 adults:
P(X=3) = C(8, 3) * (0.2353)^3 * (1-0.2353)^(8-3) = 56 * 0.0553^3 * 0.2342 ≈ 0.1373
Step 3: Calculate the probability that exactly 4 adults have been told they have hypertension.
Using the binomial probability formula once again, we can calculate the probability for exactly 4 adults:
P(X=4) = C(8, 4) * (0.2353)^4 * (1-0.2353)^(8-4) = 70 * 0.0553^4 * 0.5181 ≈ 0.0964
Step 4: Sum up the probabilities for each case.
To find the probability that between 2 and 4 (inclusive) adults have been told they have hypertension, we sum up the probabilities from steps 1 to 3:
P(2 ≤ X ≤ 4) = P(X=2) + P(X=3) + P(X=4)
= 0.1214 + 0.1373 + 0.0964
≈ 0.3551
Therefore, the probability that between 2 and 4 (inclusive) of the 8 randomly selected adults have been told they have hypertension is approximately 0.3551, or 35.51%.
To find the probability that between 2 and 4 adults (inclusive) 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 C k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials (in this case, the number of adults randomly selected)
- k is the number of successful trials (in this case, the number of adults told they have hypertension)
- p is the probability of success in a single trial (the percentage of adults who have been told they have hypertension, which is 23.53% or 0.2353)
- (n C k) represents the number of possible combinations of choosing k successes out of n trials (which can be calculated using binomial coefficients)
Now, we need to calculate the probability for each value of k between 2 and 4 (inclusive) and then sum them up.
P(2 adults have been told they have hypertension):
P(X = 2) = (8 C 2) * 0.2353^2 * (1-0.2353)^(8-2)
P(3 adults have been told they have hypertension):
P(X = 3) = (8 C 3) * 0.2353^3 * (1-0.2353)^(8-3)
P(4 adults have been told they have hypertension):
P(X = 4) = (8 C 4) * 0.2353^4 * (1-0.2353)^(8-4)
Finally, we need to sum up these individual probabilities:
P(between 2 and 4 adults inclusive have been told they have hypertension) = P(X = 2) + P(X = 3) + P(X = 4)
By calculating these probabilities using the formula and performing the necessary calculations, you can find the result.