NBC News reported on May 2, 2013, that 1 in 20 children in the United States have a food allergy of some sort. Consider selecting a random sample of 25 children and let X be the number in the sample who have a food allergy. Then X ~ Bin(25, .05).

a. Determine both P(X <= 3) and P(X < 3).

b. Determine P(X >= 4). c. Determine P(1 <= X <= 3).

d. What are E(X) and Sigma (x)?

e. In a sample of 50 children, what is the probablity that none have a food allergy?

(a) P(X<=3) = 0.9658, P(X<3) = 0.8728

(b)  P(X>=4) = 0.0342

(c) P(1<= X <=3) = 0.6884

(d)  E(X) = 1.25, σ(X) = 1.089

(e) P(X=0) = 0.0769

Step-by-step explanation:

We are given that X ~ Bin(25, .05) which means that this can be approximated as a binomial distribution. The formula for calculating probability using the binomial distribution is:

P(X=x) = ⁿCₓ pˣ qⁿ⁻ˣ

where n = total no. of trials

          x = no. of successful trials

          p = probability of success

          q = probability of failure = 1 - p

We have n = 25, p = 0.05 and q = 0.95

(a) P(X<=3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

                 = ²⁵C₀ (0.05)⁰ (0.95)²⁵⁻⁰ + ²⁵C₁ (0.05)¹ (0.95)²⁵⁻¹ + ²⁵C₂ (0.05)² (0.95)²⁵⁻² + ²⁵C₃ (0.05)³ (0.95)²⁵⁻³

                 = 0.2774 + 0.3649 + 0.2305 + 0.09302

   P(X<=3) = 0.9658

   P(X<3) = P(X=0) + P(X=1) + P(X=2)

               = ²⁵C₀ (0.05)⁰ (0.95)²⁵⁻⁰ + ²⁵C₁ (0.05)¹ (0.95)²⁵⁻¹ + ²⁵C₂ (0.05)² (0.95)²⁵⁻²

               = 0.2774 + 0.3649 + 0.2305

   P(X<3) = 0.8728

(b) P(X>=4) = 1 - P(X<4)

                   = 1 - (P(X=0) + P(X=1) + P(X=2) + P(X=3))

                  = 1 - 0.9658

    P(X>=4) = 0.0342

(c) P(1<= X <=3) = P(X=1) + P(X=2) + P(X=3)

                         = ²⁵C₁ (0.05)¹ (0.95)²⁵⁻¹ + ²⁵C₂ (0.05)² (0.95)²⁵⁻² + ²⁵C₃ (0.05)³ (0.95)²⁵⁻³

                         = 0.3649 + 0.2305 + 0.09302

   P(1<= X <=3) = 0.6884

(d) E(X) = np

            = (25)(0.05)

    E(X) = 1.25

  σ(X) = √npq

          = √(25)(0.05)(0.95)

  σ(X) = 1.089

(e) n=50 and p= 0.05. We need P(X=0) so,

P(X=0) = ⁵⁰C₀ (0.05)⁰ (0.95)⁵⁰⁻⁰

           = (1)*(1)*(0.0769)

P(X=0) = 0.0769

step-by-step explanation:

ok one would be money, and the other would be refreshments or something similar. if each bag of popcorn is 5.00 and each drink is half of that that means drinks are 2.50 the proper way to do that would to look and see how much of each you can buy without exceeding a limit of 30.00