Question

Suppose that 30% of all college students smoke cigarettes. A sample of 14 is selected randomly. What is the probability that at least 2 students smoke? Round your answer to four decimal places.

Answer 1

This is a direct application of binomial distribution.

Let X be a number of students smoke among 14 selected students.

Here, X ~ Binomial ( n = 14, p = 0.30)

Probability mass function of X is,

P(X = x) = ⁿC_x p^x (1 - p)^n-x

We want to find, P(X >= 2)

P(X >= 2)

= 1 - P(X < 2)

= 1 - [ P(X = 0) +  P(X = 1) ]

= 1 - [ 0.0068 + 0.0407 ]

= 1 - 0.0475

= 0.9525

Therefore, the probability that at least 2 students smoke is 0.9525

Note:

P(X = 0)

= ¹⁴C₀ * (0.3)⁰ * (1 - 0.3)^14-0

= 1 * 1 * (0.7)¹⁴

= 0.0068

=> P(X = 0) = 0.0068

and

P(X = 1) = ¹⁴C₁ * (0.3)¹ * (0.7)¹³ = 14 * 0.3 * (0.7)¹³ = 0.0407