Question

3. 95% of the radios produced by a company are known to have no defects. Three radios are tested at random and the number of defective radios is recorded.

a) Define and name the random variable in this problem. (1 mark)

b) Find the probability distribution for this random variable. Express all probabilities correctly rounded to 4 decimal places.

c) What is the expected number of non-defective radios?

d) Determine the probability of at most 2 defective radios.

Answer 1

a)

X ~ Bin ( n , p)

Random variable is x where x is the number of defective radios is recorded.

b)

P(X) = p = Probability of defective radios. = 1 - 0.95 = 0.05

P(X) = nCx * p^x * ( 1 - p)^n-x

P(X = 0) = ³C₀ * 0.05⁰ * ( 1 - 0.05)³ = 0.8574

P(X = 1) = ³C₁ * 0.05¹ * ( 1 - 0.05)² = 0.1354

P(X = 2) = ³C₂ * 0.05² * ( 1 - 0.05)¹ = 0.0071

P(X = 3) = ³C₃ * 0.05³ * ( 1 - 0.05)⁰ = 0.0001

c)

P(non defective) = 0.95

E(X) = n * p = 3 * 0.95

= 2.85

d)

P(X <= 2) = 1 - P(X = 3)

= 1 - 0.0001

= 0.9999