To analyze the statements, let's first define the total number of movies Christina has to choose from. She has:
- 9 action movies
- 7 science fiction movies
- 4 comedies
This gives her a total of:
\[ 9 + 7 + 4 = 20 \] movies.
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Probability of choosing three comedies:
To find the probability that Christina chooses three comedies, we need to calculate the number of ways to choose 3 comedies out of 4, relative to the number of ways to choose 3 movies from 20 total movies.
The expression for this probability would be:
\[ \frac{\binom{4}{3}}{\binom{20}{3}} = \frac{4}{1140} \text{ (which is the total ways to choose 3 from 20)} \]
The expression given in the statement, \( \frac{1}{4 C 3} \), is incorrect because it does not take into account the total number of movies when calculating the probability.
This statement is false.
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Probability of choosing three action movies:
The probability Christina will choose 3 action movies can be expressed as the number of ways to choose 3 action movies out of 9 divided by the number of ways to choose any 3 movies out of 20.
This would be:
\[ \frac{\binom{9}{3}}{\binom{20}{3}} = \frac{84}{1140} \text{ (for 9 action movies)} \]
The given statement \( \frac{20 C 3}{9 C 3} \) misrepresents the situation because it incorrectly implies choosing action movies relative to total choices.
This statement is false.
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Probability that Christina will not choose all comedies:
To find the probability that Christina does not choose all comedies, we can use the complement rule. The probability of choosing all comedies (3 comedies) is given by:
\[ \frac{\binom{4}{3}}{\binom{20}{3}}. \]
Therefore, the probability that she does not choose all comedies would be:
\[ 1 - \frac{\binom{4}{3}}{\binom{20}{3}}. \]
This does indeed align with the statement given, \( 1 - \frac{4 C 3}{20 C 3} \).
This statement is true.
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Probability that Christina will not choose all action movies:
Similarly, the probability of not choosing all action movies can be found by considering the probability of choosing all action movies (3 out of 9) compared to the total of 20 movies.
The probability of choosing 3 action movies is:
\[ \frac{\binom{9}{3}}{\binom{20}{3}}. \]
The statement given suggests that the probability of not choosing all action movies is:
\[ 1 - \frac{9 C 3}{20 C 3}. \]
This is incorrect as it doesn't correctly compare the event of not selecting all action movies against the total.
This statement is false.
Overall, the only true statement is:
The probability that Christina will not choose all comedies can be expressed as \( 1 − \frac{4 C 3}{20 C 3} \).
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