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

Answer:

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

Explanation:

Step1: Calculate total number of movies

There are $9 + 7+4=20$ movies in total.

Step2: Recall combination - probability formula

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

Step3: Analyze probability of choosing all - action movies

The number of ways to choose 3 action movies out of 9 action movies is ${9}C{3}$. So the probability of choosing all action movies is $P(\text{all action})=\frac{{9}C{3}}{{20}C{3}}$.

Step4: Analyze probability of not - choosing all action movies

The probability of the complement of an event $A$ (not - $A$) is $P(\text{not }A)=1 - P(A)$. So the probability of not choosing all action movies is $1-\frac{{9}C{3}}{{20}C{3}}$.

Step5: Analyze other options

• For choosing 3 comedies: The probability should be $\frac{{4}C{3}}{{20}C{3}}$, not $\frac{1}{{4}C{3}}$.
• For choosing 3 action movies: The probability should be $\frac{{9}C{3}}{{20}C{3}}$, not $\frac{{20}C{3}}{{9}C{3}}$.
• For not choosing all comedies: The probability should be $1-\frac{{4}C{3}}{{20}C{3}}$, not $1-\frac{{4}C{3}}{{20}C{4}}$.