Question

If x is a binomial random variable, compute P(x) for each of the following cases:

(a)  P(x≤5),n=9,p=0.7P(x≤5),n=9,p=0.7

(b)  P(x>1),n=9,p=0.1P(x>1),n=9,p=0.1

(c)  P(x<3),n=5,p=0.6P(x<3),n=5,p=0.6

(d)  P(x≥1),n=6,p=0.9P(x≥1),n=6,p=0.9

Answer 1

Solution

Given that ,

a) p = 0.7

1 - p = 1 - 0.7 = 0.3

n = 9

Using binomial probability formula ,

P(X x) = (_n C _x) * p^x * (1 - p)^{n - x}

P(X 5 ) = (₉C ₅) * 0.7⁵ * (0.3)⁴

= 0.270340

Probability = 0.2703

b) p = 0.1

1 - p = 1 - 0.1 = 0.9

n = 9

Using binomial probability formula ,

P(X > x) = (_n C _x) * p^x * (1 - p)^{n - x}

P(X > 1 ) = (₉C ₁) * 0.1¹ * (0.9)⁸

= 0.225159

Probability = 0.2252

c) p = 0.6

1 - p = 1 - 0.6 = 0.4

n = 5

Using binomial probability formula ,

P(X < x) = (_n C _x) * p^x * (1 - p)^{n - x}

P(X < 3 ) = (₅C ₃) * 0.6³ * (0.4)²

= 0.31744

Probability = 0.3174

d) p = 0.9

1 - p = 1 - 0.9 = 0.1

n = 6

Using binomial probability formula ,

P(X x) = (_n C _x) * p^x * (1 - p)^{n - x}

P(X 1 ) = (₆ C ₁) * 0.9¹ * (0.1)⁵

= 0.9999

Probability = 0.9999