Probability

Question

10.7% of the population is 65 or older. Find the probability that the following number of persons selected at random from 25 people are 65 or older.

The probability that exactly 2 are 65 or older is ...?

The probability at most 2 are 65 or older is...?

Answer 1

This is an application of the Binomial Distribution P{X=k} = C(n, k)p^k(1-p)^n-k

 

10.7% of the population is 65 or older

⇒ p = 0.107 , and 1-p = 0.893

Let X = number of persons older than 65 out of 25 selected.

 

a) The probability that exactly 2 are 65 or older is

P{X=2} = C(25, 2)p²(1-p)^25-2

             = 300(0.107)²(0.893)²³

             = 0.2543

 

b) The probability at most 2 are 65 or older is

P{X≤2} = P{X=0} + P{X=1} + P{X=2}

            = C(25,0)(0.107)⁰(0.893)²⁵ + C(25,1)(0.107)¹(0.893)²⁴ + 0.2543

            = 0.4903