Respuesta :
Answer:
a. P(1 Sports illustrated and 3 Newsweek) = [tex]\frac{100}{1001}[/tex]
b. P(at least 2 Newsweek) = [tex]\frac{595}{1001}[/tex]
We follow these steps to arrive at the answer:
The total number of magazines is [tex]5+6+3 =14[/tex]
Of these we need to choose 4 books randomly. Since order doesn’t matter here, we use combinations.
There are [tex]_{14}C_{4}[/tex] ways to choose 4 books.
[tex]_{14}C_{4}=\frac{14!}{10!4!} = 1001 ways[/tex]
1. 1. We need to choose 1 Sports illustrated and 3 newsweeks.
We can choose 1 Sports illustrated from 5 in [tex]_{5}C_{1}[/tex] ways.
[tex]_{5}C_{1}=\frac{5!}{4!1!} = 5 ways[/tex]
We can choose 3 issues of Newsweek in [tex]_{6}C_{3}[/tex] ways.
[tex]_{6}C_{3}=\frac{6!}{3!3!} = 20 ways[/tex]
So,
[tex]P(1 Sports Illustrated and 3 Newsweek) = \frac{_{6}C_{3} * _{5}C_{1}}{_{14}C_{4}}[/tex]
[tex]P(1 Sports Illustrated and 3 Newsweek) = \frac{20 * 5}{1001}[/tex]
[tex]P(1 Sports Illustrated and 3 Newsweek) = \frac{100}{1001}[/tex]
2. P(at least 2 Newsweek)
Here we divide the number of books into two categories. One is the Newsweek Magazines (6), the other group consists of all other books (8).
[tex]P(at least 2 Newsweek) = 1 - [P(no Newsweek) + P(one newsweek)][/tex]
Now, if we have to find P(no newsweek), we first need to find the number of ways we can select 4 books from the group that does NOT contain Newsweek magazines.
We can calculate that by:
[tex]{ _{8}C_{4} = \frac{8!}{4!4!} = 70 ways[/tex]
So,
[tex]P(no newsweek) = \frac{70}{1001}[/tex]
We can compute P(1 Newsweek) as follows:
We can choose 1 Newsweek in [tex]_{6}C_{1} =\frac{6!}{5!1!} = 6 ways[/tex]
We can choose 3 other books in [tex]_{8}C_{3} =\frac{8!}{5!3!} = 56 ways
So, P(at least 1 Newsweek) = [tex]\frac{6*56}{1001} or \frac{336}{1001}[/tex]
Since
[tex]P(at least 2 Newsweek) = 1 - [P(no Newsweek) + P(one newsweek)][/tex],
[tex]P(at least 2 Newsweek) = 1 - [\frac{70}{1001} + \frac{336}{1001}][/tex]
[tex]P(at least 2 Newsweek) = 1 - [\frac{406}{1001}][/tex]
[tex]P(at least 2 Newsweek) = \frac{595}{1001}[/tex]
