Christina is randomly choosing three movies to take on vacation from nine action movies, seven science fiction movies, and four comedies. Which statement is true?

A. The probability that Christina will choose three comedies can be expressed as [tex]\(\frac{1}{{4 \choose 3}}\)[/tex].

B. The probability that Christina will choose three action movies can be expressed as [tex]\(\frac{{20 \choose 3}}{{9 \choose 3}}\)[/tex].

C. The probability that Christina will not choose all comedies can be expressed as [tex]\(1 - \frac{{4 \choose 3}}{{20 \choose 4}}\)[/tex].

D. The probability that Christina will not choose all action movies can be expressed as [tex]\(1 - \frac{{9 \choose 3}}{{20 \choose 3}}\)[/tex].



Answer :

Let's analyze each statement carefully using the given probabilities:

1. Probability that Christina will choose three comedies can be expressed as [tex]$\frac{1}{{ }_4 C_3}$[/tex]:
- There are 4 comedy movies, and the number of ways to choose 3 out of these 4 is denoted by [tex]\({ }_4 C_3\)[/tex].
- The formula for the probability is [tex]\(\frac{1}{{ }_4 C_3}\)[/tex].
- According to the result, the probability is 0.25.
- Since [tex]\(\frac{1}{4} = 0.25\)[/tex], this statement matches our calculated probability.
- This statement is true.

2. Probability that Christina will choose three action movies can be expressed as [tex]$\frac{{ }_{20} C_3}{9 C_3}$[/tex]:
- Total number of action movies is 9, and the number of ways to choose 3 out of these 9 is denoted by [tex]\({ }_9 C_3\)[/tex].
- Total movies available are 20, and the number of ways to choose 3 out of these 20 is denoted by [tex]\({ }_{20} C_3\)[/tex].
- The formula for this probability is [tex]\(\frac{{ }_9 C_3}{{ }_{20} C_3}\)[/tex].
- The given form [tex]\(\frac{{ }_{20} C_3}{{ }_9 C_3}\)[/tex] would actually be the inverse of what we need.
- This statement is false.

3. Probability that Christina will not choose all comedies can be expressed as [tex]\(1 - \frac{{ }_{4} C_{3}}{ { }_{20} C_{4}}\)[/tex]:
- The total number of ways to choose 4 movies out of 20 is denoted by [tex]\({ }_{20} C_4\)[/tex].
- The number of ways to choose 3 comedies out of 4 is denoted by [tex]\({ }_4 C_3\)[/tex].
- According to our probabilities, the formula for not choosing all comedies is [tex]\(1\)[/tex] minus the probability of choosing all comedies (given by [tex]\(\frac{{ }_{4} C_{3}}{ { }_{20} C_{4}}\)[/tex]).
- The calculated probability is approximately 0.999174, which matches the form given.
- This statement is true.

4. Probability that Christina will not choose all action movies can be expressed as [tex]\(1 - \frac{{ }_9 C_3}{ { }_{20} C_3}\)[/tex]:
- As per the previous discussions, the probability of choosing 3 action movies from 9 out of total 20 movies is [tex]\(\frac{{ }_9 C_3}{{ }_{20} C_3}\)[/tex].
- To find the probability of not choosing all action movies, we take [tex]\(1\)[/tex] minus this probability.
- The calculated probability is approximately 0.926316, matching the form of [tex]\(1 - \frac{{ }_9 C_3}{{ }_{20} C_3}\)[/tex].
- This statement is true.

In conclusion, statements 1, 3, and 4 are true, while statement 2 is false.