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?\no the probability that christina will choose three comedies can be expressed as $\frac{1}{_{4}c_{3}}$.\no the probability that christina will choose three action movies can be expressed as $\frac{_{20}c_{3}}{_{9}c_{3}}$.\no the probability that christina will not choose all comedies can be expressed as $1 - \frac{_{4}c_{3}}{_{20}c_{4}}$.\no 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 movies, (7) science - fiction movies and (4) comedies. So the total number of movies is (9 + 7+4=20) movies.
Step2: Recall probability formula for combinations
The probability of an event (E) is (P(E)=\frac{n(E)}{n(S)}), where (n(E)) is the number of elements in the event set and (n(S)) is the number of elements in the sample - space. The number of ways to choose (r) elements from (n) elements is given by the combination formula ({n}C{r}=\frac{n!}{r!(n - r)!}). The sample - space (n(S)) (the number of ways to choose (3) movies out of (20)) is ({20}C{3}).
Step3: Analyze probability of choosing all comedies
The number of ways to choose (3) comedies out of (4) is ({4}C{3}). So the probability of choosing (3) comedies is (P(\text{3 comedies})=\frac{{4}C{3}}{{20}C{3}}).
Step4: Analyze probability of choosing all action movies
The number of ways to choose (3) action movies out of (9) is ({9}C{3}). So the probability of choosing (3) action movies is (P(\text{3 action movies})=\frac{{9}C{3}}{{20}C{3}}).
Step5: Analyze probability of not choosing all comedies
The probability of choosing all comedies is (P(\text{3 comedies})=\frac{{4}C{3}}{{20}C{3}}). The probability of not choosing all comedies is (P(\text{not 3 comedies}) = 1-\frac{{4}C{3}}{{20}C{3}}).
Step6: Analyze probability of not choosing all action movies
The probability of choosing all action movies is (P(\text{3 action movies})=\frac{{9}C{3}}{{20}C{3}}). The probability of not choosing all action movies is (P(\text{not 3 action movies})=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}}).