A bookshelf has four identical-looking books which are 200, 400, 600, and 800 pages long. Velma picks a random book off the shelf, flips to a random page to read, and puts the book back on the shelf. Later, Daphne also picks a random book off the shelf and flips to a random page to read. Given that Velma read page 122 of her book and Daphne read page 304 of her book, the probability that they chose the same book is m n for relatively prime positive integers m and n. Compute 100m + n.

316

Step-by-step explanation:

The question has a little problem:

"the probability that they chose the same book is m n for relatively prime positive integers m and n. Compute 100m + n."

The correct sentence:

the probability that they chose the same book is "m/n" for relatively prime positive integers m and n.

Total number of books = 4

We have:

Number of 200page book = 1

Number of 400page book = 1

Number of 600page book = 1

Number of 800page book = 1

Probability of picking same book:

Velma read page 122 of her book Daphne read page 304 of her book

If it is same book, it must contain atleast 400page.

Therefore, 400page, 600page and 800 page would be considered in the probability.

Pr(one 400page) = 1/4

Pr(picking two 400page) = 1/4 * 1/4 = 1/16

Pr(one 600page) = 1/4

Pr(picking two 600page) = 1/4 * 1/4 = 1/16

Pr(one 800page) = 1/4

Pr(picking two 800page) = 1/4 * 1/4 = 1/16

Pr(picking same book page)=

Pr(picking two 400page) or Pr(picking two 600page) or Pr(picking two 800page)

= Pr(picking two 400page) + Pr(picking two 600page) + Pr(picking two 800page) = 1/16+ 1/16+ 1/16

Pr(picking same book page)= 3/16

This answer satisfies the probability as m/n for relatively prime positive integers m and n.

Two numbers are said to be relatively prime integers if the only positive integer that divides both of them is 1. It means the numerator and denominator of the fraction have been reduced to the lowest form.

m/n = 3/16

m = 3, n= 16

100m + n = 100(3) + 16

= 316