Question

In your voting district, 35% of the voters are against a particular bill and the rest favor it. If you randomly poll four voters from your district, what is the probability that

(a) None will favor the bill?
  

(b) All will favor the bill?
  

(c) At least two will be against the bill?

Answer 1

a)
Here, n = 4, p = 0.65, (1 - p) = 0.35 and x = 0
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 0)
P(X = 0) = 4C0 * 0.65^0 * 0.35^4
P(X = 0) = 0.015

b)
We need to calculate P(X = 4)
P(X = 4) = 4C4 * 0.65^4 * 0.35^0
P(X = 4) = 0.1785

c)
Here, n = 4, p = 0.35, (1 - p) = 0.65 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 2).
P(X >= 2) = (4C2 * 0.35^2 * 0.65^2) + (4C3 * 0.35^3 * 0.65^1) + (4C4 * 0.35^4 * 0.65^0)
P(X >= 2) = 0.3105 + 0.1115 + 0.015
P(X >= 2) = 0.437