Answered

Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Mariah is randomly choosing three books to read from the following: 5 mysteries, 7 biographies, and 8 science fiction novels.

Which of these statements are true? Check all that apply.

A. There are [tex]\({ }_{20} C _3\)[/tex] possible ways to choose three books to read.
B. There are [tex]\({ }_5 C _3\)[/tex] possible ways to choose three mysteries to read.
C. There are [tex]\({ }_{15} C _3\)[/tex] possible ways to choose three books that are not all mysteries.
D. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{ }_5 C_3}\)[/tex].
E. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{ }_5 C_3}{{ }_{20} C_3}\)[/tex].

Sagot :

GinnyAnswer
Let's break down each statement one by one and analyze whether it is true based on the calculations:

1. There are [tex]\({ }_{20} C _3\)[/tex] possible ways to choose three books to read.

We need to find the total number of combinations of choosing 3 books from 20. The formula for combinations is given by [tex]\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)[/tex].

Here, we need to find [tex]\(\binom{20}{3}\)[/tex]. This evaluates to 1140.

Since the calculation aligns with the statement, this statement is true.

2. There are [tex]\({ }_5 C _3\)[/tex] possible ways to choose three mysteries to read.

The number of ways to choose 3 mysteries from 5 mysteries is given by [tex]\(\binom{5}{3}\)[/tex].

This evaluates to 10.

Since this is correct, the statement is true.

3. There are [tex]\({ }_{15} C _3\)[/tex] possible ways to choose three books that are not all mysteries.

Here, we need to find the number of combinations of choosing 3 books from the total 15 non-mystery books (7 biographies + 8 science fiction).

We calculate [tex]\(\binom{15}{3}\)[/tex]. This evaluates to 455.

The statement is confirmed to be true.

4. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{\binom{5}{3}}\)[/tex].

To find the probability of choosing 3 mysteries, we divide the number of ways to choose 3 mysteries by the total number of ways to choose 3 books.

[tex]\(\text{Probability} = \frac{\binom{5}{3}}{\binom{20}{3}} = \frac{10}{1140} = 0.008771929824561403\)[/tex].

This does not align with [tex]\(\frac{1}{\binom{5}{3}}\)[/tex], which would be [tex]\(\frac{1}{10} = 0.1\)[/tex].

Therefore, this statement is false.

5. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{\binom{5}{3}}{\binom{20}{3}}\)[/tex].

First, we calculate the probability of choosing all mysteries, which is [tex]\(\frac{\binom{5}{3}}{\binom{20}{3}} = \frac{10}{1140} = 0.008771929824561403\)[/tex].

Then, the probability of not choosing all mysteries is:
[tex]\[ 1 - \text{Probability of choosing all mysteries} = 1 - 0.008771929824561403 = 0.9912280701754386 \][/tex]

Since this matches our results, the statement is true.

Summarizing, the statements:
1, 2, 3, and 5 are true.
Statement 4 is false.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.