Answer:
a) 0.294
b) 0.414
c) 0.694
Step-by-step 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
