Question

23% of U.S adults say they are more likely to make purchases during a sales tax holiday. You randomly select 10 adults. Find the probability that the number of adults who say they are more likely to make purchases during a sales tax holiday is a) exactly two, b) more than two, and C) between two and five, inclusive

Answer 1

a) 0.294

b) 0.414

c) 0.694

Explanation:

Use binomial probability.

P = nCr pʳ qⁿ⁻ʳ

where n is the number of trials,

r is the number of successes,

p is the probability of success,

and q is the probability of failure (1−p).

In this case, n = 10, p = 0.23, and q = 0.77.

a) If r = 2, then the probability is:

P = ₁₀C₂ (0.23)² (0.77)¹⁰⁻²

P = 45 (0.23)² (0.77)⁸

P = 0.294

b) If r > 2, then the probability is 1 − P(r ≤ 2).

P = 1 − ₁₀C₀ (0.23)⁰ (0.77)¹⁰⁻⁰ − ₁₀C₁ (0.23)¹ (0.77)¹⁰⁻¹ − ₁₀C₂ (0.23)² (0.77)¹⁰⁻²

P = 1 − 1 (0.23)⁰ (0.77)¹⁰ − 10 (0.23)¹ (0.77)⁹ − 45 (0.23)² (0.77)⁸

P = 1 − 0.073 − 0.219 − 0.294

P = 0.414

c) If 2 ≤ r ≤ 5, then the probability is:

P = ₁₀C₂ (0.23)² (0.77)¹⁰⁻² + ₁₀C₃ (0.23)³ (0.77)¹⁰⁻³ + ₁₀C₄ (0.23)⁴ (0.77)¹⁰⁻⁴ + ₁₀C₅ (0.23)⁵ (0.77)¹⁰⁻⁵

P = 45 (0.23)² (0.77)⁸ + 120 (0.23)³ (0.77)⁷ + 210 (0.23)⁴ (0.77)⁶ + 252 (0.23)⁵ (0.77)⁵

P = 0.294 + 0.234 + 0.122 + 0.044

P = 0.694