Question

Problem 7-22 A food-packaging apparatus underfills 20% of the containers. Find the probability that for any particular 10 containers the number of underfilled will be: Note: This question uses a tolerance of .0001 for grading. Make sure your answers are entered to the fifth decimal place. Example: .00123

Answer 1

(a) exactly 2

P(X=2) = 0.30199

(b) exactly 4

P(X=4) = 0.08808

(c) exactly 10

P(X=10) ≈ 0.00000

(d) at least 5

P(X ≥ 5) = 0.03279

Explanation:

This is a binomial distribution problem

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = number of containers to be considered = 10

x = Number of successes required = number of underfilled containers required

p = probability of success = probability that a container is underfilled = 0.20

q = probability of failure = probability that a container is NOT underfilled = 1 - 0.20 = 0.80

a) x = 2

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

P(X = 2) = ¹⁰C₂ (0.2)² (0.8)¹⁰⁻² = 0.30199

b) x = 4

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

P(X = 4) = ¹⁰C₄ (0.2)⁴ (0.8)¹⁰⁻⁴ = 0.08808

c) x = 10

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

P(X = 2) = ¹⁰C₁₀ (0.2)¹⁰ (0.8)¹⁰⁻¹⁰ ≈ 0.00000

d) x ≥ 5

P(X ≥ 5) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

Solving for each of these probabilities and summing them all up,

P(X ≥ 5) = P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) = 0.0327934976 = 0.03279

Hope this Helps!!!