christina is randomly choosing three movies to take on vacation from nine action movies, seven science…

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? the probability that christina will choose three comedies can be expressed as $\frac{1}{_{4}c_{3}}$. the probability that christina will choose three action movies can be expressed as $\frac{_{20}c_{3}}{_{9}c_{3}}$. the probability that christina will not choose all comedies can be expressed as $1 - \frac{_{4}c_{3}}{_{20}c_{4}}$. the probability that christina will not choose all action movies can be expressed as $1 - \frac{_{9}c_{3}}{_{20}c_{3}}$.

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? the probability that christina will choose three comedies can be expressed as $\frac{1}{_{4}c_{3}}$. the probability that christina will choose three action movies can be expressed as $\frac{_{20}c_{3}}{_{9}c_{3}}$. the probability that christina will not choose all comedies can be expressed as $1 - \frac{_{4}c_{3}}{_{20}c_{4}}$. the probability that christina will not choose all action movies can be expressed as $1 - \frac{_{9}c_{3}}{_{20}c_{3}}$.

Answer

Explanation:

Step1: Calculate total number of movies

There are (9) action, (7) science - fiction and (4) comedies. So total movies (n=9 + 7+4=20).

Step2: Recall probability formula for combinations

The probability of an event (E) when choosing (r) items out of (n) is (P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}). The total number of ways to choose (3) movies out of (20) is ({20}C{3}=\frac{20!}{3!(20 - 3)!}).

Step3: Analyze probability of choosing 3 comedies

The number of ways to choose (3) comedies out of (4) is ({4}C{3}=\frac{4!}{3!(4 - 3)!}=4). The probability of choosing 3 comedies is (\frac{{4}C{3}}{{20}C{3}}), not (\frac{1}{{4}C{3}}).

Step4: Analyze probability of choosing 3 action movies

The number of ways to choose 3 action movies out of 9 is ({9}C{3}=\frac{9!}{3!(9 - 3)!}). The probability of choosing 3 action movies is (\frac{{9}C{3}}{{20}C{3}}), not (\frac{{20}C{3}}{{9}C{3}}).

Step5: Analyze probability of not choosing all comedies

The probability of choosing all comedies is (\frac{{4}C{3}}{{20}C{3}}). The probability of not choosing all comedies is (1-\frac{{4}C{3}}{{20}C{3}}), not (1 - \frac{{4}C{3}}{{20}C{4}}).

Step6: Analyze probability of not choosing all action movies

The probability of choosing all action movies is (\frac{{9}C{3}}{{20}C{3}}). The probability of not choosing all action movies is (1-\frac{{9}C{3}}{{20}C{3}}).

Answer:

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