Question

Market research shows that 27% of college students have at least one major credit card. If 10 students are selected at random, calculate the probability that fewer than 2 of them have at least one major credit card. What about the probability that exactly 5 out of the 10 students have at least one major credit card? Find the mean and standard deviation of this probability distribution.

Answer 1

Solution

Given that ,

p = 0.27

1 - p = 0.73

n = 10

Using binomial probability formula ,

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

1)

P(X< 2) = P(X = 0 ) + P(X = 1 )

= ((10! / 0! (10-0)!) * 0.27⁰ * (0.73)^10-0 + ((10! / 1! (10-1)!) * 0.27¹ * (0.73)^10-1

=  ((10! / 0! (10)!) * 0.27⁰ * (0.73)⁰ + ((10! / 1! (9)!) * 0.27¹ * (0.73)¹

= 0.0430 + 0.1590

= 0.2020

Probability =

2)

P(X = 5) = ((10! / 5! (10-5)!) * 0.27⁵ * (0.73)^10-5

⁼ ((10! / 5! (5)!) * 0.27⁵ * (0.73)⁵

= 0.0750

Probability =

3)

Mean = = n*P = 10*0.27 = 2.7

Standard deviation = =n*p*(1-p) = 10*0.27*0.73 = 1.971 = 1.40