Question

Assume that X is a binomial random variable with n = 15 and p = 0.78. Calculate the following probabilities. (Do not round intermediate calculations. Round your final answers to 4 decimal places.)

Assume that X is a binomial random variable with n = 15 and p = 0.78. Calculate the following probabilities. (Do not round intermediate calculations. Round your final answers to 4 decimal places.)

a.	P(X = 14)
b.	P(X = 13)
c.	P(X ≥ 13)

Answer 1

Solution

Given that ,

p = 0.78

1 - p = 1 - 0.78 = 0.22

n = 15

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

a)

x = 14

P(X = 14) = ((15! / 14! (1)!) * 0.78¹⁴ * (0.22)¹

= 0.1018

Probability = 0.1018

b)

x = 13

P(X = 13) = ((15! / 13! (2)!) * 0.78¹³ * (0.22)²

= 0.2010

Probability = 0.2010

b)

P(X 13) = P(X = 13) + P(X = 14) + P(X = 15)

=  ((15! / 13! (2)!) * 0.78¹³ * (0.22)² + ((15! / 14! (1)!) * 0.78¹⁴ * (0.22)¹ + ((15! / 15! (0)!) * 0.78¹⁵ * (0.22)⁰

= 0.3269

Probability = 0.3269