In a study, 35% of adults questioned reported that their health was excellent. A researcher wishes to study health of people living close to a nuclear power plant. Among 14 adults randomly selected from this area, only 3 reported that their health was excellent. Find the probability that when 14 adults are randomly selected, 3 or fewer are in excellent health. SHOW WORK/FORMULA USED OR STATE CALCULATOR STEP USED
To find the probability that 3 or fewer adults out of 14 are in excellent health, we need to use the binomial distribution formula. The formula for the probability mass function of the binomial distribution is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes.
- n is the number of trials or individuals selected.
- k is the number of successes.
- p is the probability of a single trial being a success.
In this case, the probability of an adult having excellent health is 35%, which is equivalent to 0.35. Therefore, p = 0.35.
We want to find the probability of having 3 or fewer adults in excellent health out of 14, which means we need to calculate the following probabilities:
P(X = 0), P(X = 1), P(X = 2), and P(X = 3)
To calculate these probabilities, we can use the binomial distribution formula for each value of k and add them together.
P(X = 0) = C(14, 0) * 0.35^0 * (1 - 0.35)^(14 - 0)
P(X = 1) = C(14, 1) * 0.35^1 * (1 - 0.35)^(14 - 1)
P(X = 2) = C(14, 2) * 0.35^2 * (1 - 0.35)^(14 - 2)
P(X = 3) = C(14, 3) * 0.35^3 * (1 - 0.35)^(14 - 3)
Now, let's calculate each probability:
P(X = 0) = 1 * 1 * 0.65^14 ≈ 0.088
P(X = 1) = 14 * 0.35 * 0.65^13 ≈ 0.250
P(X = 2) = 91 * 0.35^2 * 0.65^12 ≈ 0.324
P(X = 3) = 364 * 0.35^3 * 0.65^11 ≈ 0.236
Finally, we can add these probabilities together to find the desired probability:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
≈ 0.088 + 0.250 + 0.324 + 0.236
≈ 0.898
Therefore, the probability that 3 or fewer adults out of 14 living close to the nuclear power plant report excellent health is approximately 0.898.