Question

The random variable X has a binomial distribution with n = 10 and p = 0.04. Determine the following probabilities.

Round your answers to six decimal places (e.g. 98.765432).

(a) P(X=5)=Enter your answer in accordance to the item a) of the question statement

(b) P(X≤2)=Enter your answer in accordance to the item b) of the question statement

(c) P(X≥9)=Enter your answer in accordance to the item c) of the question statement

(d) P(3≤X<5)=Enter your answer in accordance to the item d) of the question statement

Answer 1

Solution:

Using binomial probability formula ,

P(X = x) = (_n C _x) * p^x * (1 - p)^{n - x} ; x = 0 ,1 , 2 , ....., n

Here , n = 10 , p = 0.04

1 - p = 1 - 0.04 = 0.96

a)

P(X = 5) = (₁₀ C ₅) * 0.04⁵ * (0.96)^{10 - 5} = 0.000021

P(X = 5) = 0.000021

b)

P(X≤2)

= P(X = 0) + P(X = 1) + P(X = 2)

= (₁₀ C₀) * 0.04⁰ * (0.96)^{10 - 0} +   (₁₀ C ₁) * 0.04¹ * (0.96)^{10 - 1} +   (₁₀ C ₂) * 0.04² * (0.96)^{10 - 2}  

=0.66483263599 + 0.27701359833 + 0.05194004969

= 0.993786

P(X≤2) = 0.993786

c)

P(X≥9)

= P(X = 9) + P(X = 10)

= 0.000000 + 0.000000

= 0.000000

P(X≥9)= 0.000000

d)

P(3≤X<5)

= P(X = 3) + P( X = 4)

= (₁₀ C₃) * 0.04³ * (0.96)^{10 - 3} +   (₁₀ C ₄) * 0.04⁴ * (0.96)^{10 - 4}

= 0.00577111663 + 0.00042081059

= 0.006192

P(3≤X<5) = 0.006192