Question

If X follows a binomial(5, 0.15), determine

(a) P(X = 0).

(b) P(X ≤ 4).

(c) P(X > 1).

Answer 1

Binomial Probability = ⁿC_x * (p)^x * (1-p)^n-x, where n = number of trials and x is the number of successes

Please note ⁿC_x = n! / [(n-x)!*x!]

Also sum of probabilities from 0 till n = 1, i.eP(0) + P(1) + P(2) +.......+P(n) = 1

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Given n = 5, p = 0.15, therefore q = 1 - p = 1 - 0.15 = 0.85

(a) P(X = 0) = ⁵C₀ * (0.15)⁰ * (0.85)⁵ = 0.4437

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(b) P(X 4) = P(4) + P(3) + P(2) + P(1) + P(0)

But Since the sum of probabilities P(5) + P(4) + P(3) + P(2) + P(1) + P(0) = 1

Therefore P(4) + P(3) + P(2) + P(1) + P(0) = 1 - P(5)

P(X = 5) = ⁵C₅ * (0.15)⁵ * (0.85)⁰ = 0.00008

Therefore P(X 4) = 1 - 0.00008 = 0.99992

(If rounded to 4 decimal places, then 1.0000)

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(c) P(X > 1) = P(2) + P(3) + P(4) + P(5) = 1 - [P(0) + P(1)]. We know from (a) P(0) = 0.4437

P(X = 1) = ⁵C₁ * (0.15)¹ * (0.85)⁴ = 0.3915

Therefore P(X > 1) = 1 - [0.4437 + 0.3915] = 1 - 0.8352 = 0.1648

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