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.
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.
