7) Billy has 13 books, but he only has space for 6 on his bookshelf. How many di

Question

7) Billy has 13 books, but he only has space for 6 on his bookshelf. How many different arrangements (order does matter) of the books can Billy make? (2 point) 8) Maggie is applying to college. There are only two outcomes for each application. It is either accepted or rejected. Each application she sends has a 0.70 probability of being accepted. She applies to 10 colleges. (6 points) (this is a binomial distribution) What is the expected value and variance for the random variable "number of acceptances"? What is the probability that she will be accepted to 3 colleges? What is the probability she will be accepted to 8 or more colleges?

Explanation / Answer

7)

Since order does matter, the number of arrangement that 13 books can be placed on 6 spaces is nPr

nPr = n! / (n-r)!

Here n = 13, r = 6

13P6 = 13! / (13-6)!

= 1235520 arrangements

8)

X ~ Bin(n,p) = Bin(10,0.7)

Expected value = np = 10 * 0.7 = 7

Variance = np(1-p) = 10 * 0.7 * 0.3 = 2.1

Binomial probability distribution is

P(X = x) = nCx px (1-p)n-x

We have to find P(X =3)

P(X = 3 ) = 10C3 0.73 0.37

= 0.0090

P(X = 3 ) = 0.0090

P(X >= 8) = P( X = 8) + P(X = 9) + P(X = 10)

= 10C8 0.78 0.32 + 10C9 0.79 0.3 + 10C10 0.710  0.30

= 0.3828

P(X >= 8) = 0.3828

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