Question

Consider a binomial probability distribution with

p= 0.65 and n=7

.

Determine the probabilities below.

​a)	Upper P left parenthesis x equals 2 right parenthesis

​b)	Upper P left parenthesis x less than or equals 1 right parenthesis
​c)	Upper P left parenthesis x greater than 5 right parenthesis

​a) Upper P left parenthesis x equals 2 right parenthesis

=   ​(Round to four decimal places as​ needed.)

Answer 1

Let X be a Binomial random variable with parameter n = 7 and p = 0.65

X ~ Binomial ( n = 7, p = 0.65)

Probability mass function of X is,

P(X = x) = ⁿC_x p^x (1 - p)^n-x

We want to find, the following probabilities,

a) P(X = 2) = ⁷C₂ * (0.65)² * (1 - 0.65)^7-2

P(X = 2) = 21 * (0.65)² * (0.35)⁵

P(X = 2) = 0.0466

b) P(X <= 1) = P(X = 0) + P(X = 1)

P(X <= 1) = [ ⁷C₀ * (0.65)⁰ * (0.35)⁷ ] + [ ⁷C₁ * (0.65)¹ * (0.35)⁶ ]

P(X <= 1) = 0.0006 + 0.0084

P(X <= 1) = 0.0090

c) P(X > 5) = P(X = 6) + P(X = 7)

P(X > 5) = [ ⁷C₆ * (0.65)⁶ * (0.35)¹ ] + [ ⁷C₇ * (0.65)⁷ * (0.35)⁰ ]

P(X > 5) = 0.1848 + 0.0490

P(X > 5) = 0.2338