Question

Sample who have a food allergy. Then X∼Bin(25,0.05). (Round your probabilities to three decimal places.)

(a) Determine both P(X≤2) and P(X<2).
P(X≤2)=
P(X<2)=
​
(b) Determine P(x≥3). P(X≥3)=
(c) Determine P(1≤X≤2). P(1≤X≤2)=
(d) What are E(X) and σ χ ? (Round your answers to two decimal places.)
E(X)=
σ X =
​ (e) In a sample of 40 children, what is the probability that none has a food allergy? You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

To solve this problem, we need to use the binomial probability distribution. Given that X follows a binomial distribution with parameters n = 25 and p = 0.05, we can calculate the probabilities as well as the expected value and standard deviation. Additionally, we can use the binomial distribution with n = 40 and p = 0.05 to find the probability that none of the children have a food allergy.

Step-by-step explanation:

To solve this problem, we need to use the binomial probability distribution. Given that X follows a binomial distribution with parameters n = 25 and p = 0.05, we can calculate the probabilities as follows:

1. P(X ≤ 2) = P(X < 2) = P(X = 0) + P(X = 1) + P(X = 2)
2. P(X ≥ 3) = 1 - P(X < 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2)
3. P(1 ≤ X ≤ 2) = P(X = 1) + P(X = 2)
4. E(X) = np = 25 * 0.05
5. σX = sqrt(np(1-p)) = sqrt(25 * 0.05 * (1-0.05))
6. To find the probability that none of the 40 children have a food allergy, we can use the binomial probability distribution again with n = 40 and p = 0.05. P(X = 0) is the probability that none of the children have a food allergy.