Question 53·Medium·Probability and Conditional Probability
A charity receives a shipment of 480 donated books and classifies each as fiction, nonfiction, or children’s. If one book is selected at random from the shipment, the probability it is classified as fiction is and the probability it is classified as nonfiction is .
How many of the donated books are classified as children’s?
(Express the answer as an integer)
For questions where you’re given probabilities for several categories that cover all possibilities, first use the fact that their probabilities must add up to 1 to find any missing probability. Then, if the problem gives a total number of items, multiply that total by the relevant probability to convert it into an actual count. Doing these two steps—“find missing probability” and then “multiply by total”—keeps the work organized and fast.
Hints
Think about all possible categories
Every book must be either fiction, nonfiction, or children’s. What should the sum of the three probabilities be?
Find what’s left over
You know the probabilities for fiction and nonfiction. Subtract their sum from 1 to get the probability of a children’s book: compute .
Turn probability into a count
Once you know the probability for children’s books, how can you use the total of 480 books to find how many are children’s? Think about what operation links a total and a probability.
Desmos Guide
Use Desmos to compute the children’s count
In a new expression line, type:
480*(1 - 0.55 - 0.30)
Look at the numerical result Desmos shows; that value is the number of children’s books in the shipment.
Step-by-step Explanation
Use the fact that probabilities add to 1
All books are classified as fiction, nonfiction, or children’s, with no overlap. That means the three probabilities must add up to 1:
You are given and .
Find the probability of a children’s book
Solve for by subtracting the known probabilities from 1:
Compute this value to get the probability that a randomly chosen book is a children’s book.
Convert the probability to a number of books
Once you have , multiply it by the total number of books, 480:
So, 72 of the donated books are classified as children’s.