Question

# The random variable X has a binomial distribution with n = 10 and p = 0.04....

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

(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

Solution:

Using binomial probability formula ,

P(X = x) = (n C x) * px * (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