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

Answer:

a.  $\frac{3}{8}$

b.  $\frac{7}{52}$

c.  $\frac{95}{156}$

Step-by-step 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,

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

$=\frac{^{15}C_1}{^{40}C_1}$

$=\frac{15}{40}$

$=\frac{3}{8}$

b. If two book is randomly selected,

Then the probability that both books are from history,

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

$=\frac{^{15}C_2}{^{40}C_2}$

$=\frac{\frac{15!}{2!13!}}{\frac{40!}{2!38!}}$

$=\frac{105}{780}$

$=\frac{7}{52}$

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,

$=^{20}C_1\times ^{15}C_1+^{20}C_1\times ^{5}C_1+^{15}C_1\times ^{5}C_1$

$=20\times 15 + 20\times 5 + 15\times 5$

$=300 + 100 + 75$

$=475$

Then the probability that both books are from different subjects

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

$=\frac{475}{780}$

$=\frac{95}{156}$