Question

Consider a binomial probability distribution with p=0.65 and n=6. Determine the probabilities below. Round to four decimal places as needeed.

a) P(x=2)

b) P(x< or equal to1)

c) P(x>4)

Answer 1

Solution

Given that ,

p = 0.65

1 - p = 1 - 0.65 = 0.35

n = 6

Using binomial probability formula ,

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

(a)

x = 2

P(X = 2) = ((6! / 2! (6 - 2)!) * 0.65² * (0.35)^{6 - 2}

=  ((6! / 2! (4)!) * 0.65² * (0.35)⁴

P(X = 2) =  0.0951

Probability = 0.0951

(b)

P(x 1) = P(x = 0) + P(x = 1)

=  ((6! / 0! (6)!) * 0.65⁰ * (0.35)⁶ +  ((6! / 1! (5)!) * 0.65¹ * (0.35)⁵

= 0.0223

P(x 1) = 0.0223

(c)

P(x > 4) = 1 - P(x 4)

= 1 - [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4)]

= 1 - [((6! / 0! (6)!) * 0.65⁰ * (0.35)⁶ +  ((6! / 1! (5)!) * 0.65¹ * (0.35)⁵ + ((6! / 2! (4)!) * 0.65² * (0.35)⁴

+ ((6! / 3! (3)!) * 0.65³ * (0.35)³ + ((6! / 4! (2)!) * 0.65⁴ * (0.35)² ]

= 1 - 0.6809

= 0.3191

P(x > 4) = 0.3191