Question

(1 point) If X is a binomial random variable, compute the probabilities for each of the following cases: (a) P(X < 1), n = 7, p = 0.3 Probability = (b) P(X > 5), n = 7, p = 0.1 Probability = (C) P(X < 6), n = 8, p = 0.5 Probability = (d) P(X > 2), n = 3, p = 0.5 Probability =

Answer 1

X ~ Bin (n,p)

P(X) = ⁿC_x * p^x ( 1 - p)^n-x

a)

P(X <= 1) = P(X = 0) + P(X = 1 )

= ⁷C₀ * 0.3⁰ * (1 - 0.3)⁷ + ⁷C₁ * 0.3¹ * (1 - 0.3)⁶

= 0.3294

b)

P(X > 5) = P(X = 6) + P(X = 7)

= ⁷C₆ * 0.1⁶ * (1 - 0.1)¹ + ⁷C₇ * 0.1⁷ * (1 - 0.1)⁰

= 0.0000

c)

P( X < 6) = 1 - P(X >= 6)

= 1 - [ P(X = 6) + P(X = 7) + P(X = 8) ]

= 1 -[ ⁸C₆ * 0.5⁶ * 0.5² + ⁸C₇ * 0.5⁷ * 0.5¹ + ⁸C₈ * 0.5⁸ * 0.5⁰ ]

= 0.8555

d)

P(X >= 2) = P(X = 2) + P(X = 3)

= ³C₂ * 0.5² * 0.5¹ + ³C₃ * 0.5³ * 0.5⁰

= 0.5