Question

1) In testing a new drug, researchers found that 5% of all patients using it will have a mild side effect. A random sample of 7 patients using the drug is selected. (i) Find the probability that none will have this mild side effect. (ii) Find the probability that at least one will have this mild side effect.

2)Given:

P(E) = 0.46, P(F) = 0.53, and P(E ∪ F) = 0.68

(i) Find P(E ∩ F)
(ii) Find P(F | E)

Answer 1

1)

X ~ binomial (n,p)

Where n = 7 , p = 0.05

Binomial probability distribution is

P(X) = ⁿC_x * p^x ( 1 - p)^n-x

a)

P( X = 0) = ⁷C₀ * 0.05⁰ * ( 1 - 0.05)⁷

= 0.6983

b)

P( X >= 1) = 1 - P( X = 0)

= 1 - ⁷C₀ * 0.05⁰ * ( 1 - 0.05)⁷

= 1 - 0.6983

= 0.3017

2)

i)

P(E F) = P(E) + P(F) - P(E F)

0.68 = 0.46 + 0.53 - P(E F)

So,

P(E F) = 0.46 + 0.53 - 0.68

= 0.31

ii)

P(F | E) = P(E F) / P(E)

= 0.31 / 0.46

= 0.6739