Question

You have 40 different books (20 math books, 15 history books, and 5 geography books). a. You pick one book at random. What is the probability that the book is a history book? b. You pick two books at random, one at a time. What is the probability that both books are history books? c. You pick two books at random, one at a time. What is the probability that the two books are from different disciplines?

Answer 1

a.
`(3)/(8)`

b.
`(7)/(52)`

c.
`(95)/(156)`

Explanation:

Given,

Total books = 40,

Maths books = 20,

History books = 15,

Geography books = 5,

a. If a book is randomly selected,

Then the probability that the book is a history book,

`=\frac{\text{Ways of choosing history book}}{\text{Ways of choosing any one book}}`

`=(^(15)C_1)/(^(40)C_1)`

`=(15)/(40)`

`=(3)/(8)`

b. If two book is randomly selected,

Then the probability that both books are from history,

`=\frac{\text{Ways of choosing two history book}}{\text{Ways of choosing any two book}}`

`=(^(15)C_2)/(^(40)C_2)`

`=((15!)/(2!13!))/((40!)/(2!38!))`

`=(105)/(780)`

`=(7)/(52)`

c. If two book is randomly selected,

Ways of selecting any two books from different subjects

= maths and history + maths and geography + history and geography,

`=^(20)C_1* ^(15)C_1+^(20)C_1* ^(5)C_1+^(15)C_1* ^(5)C_1`

`=20* 15 + 20* 5 + 15* 5`

`=300 + 100 + 75`

`=475`

Then the probability that both books are from different subjects

`=\frac{\text{Ways of selecting any two books from different subjects}}{\text{Ways of choosing any two book}}`

`=(475)/(780)`

`=(95)/(156)`