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?
A. The probability that Christina will not choose all comedies can be expressed as $1 - \frac{C_4^3}{{}_{20}C_3}$.
B. The probability that Christina will choose three action movies can be expressed as $\frac{C_9^3}{{}_{20}C_3}$.
C. The probability that Christina will not choose all action movies can be expressed as $1 - \frac{C_9^3}{{}_{20}C_3}$.
D. The probability that Christina will choose three comedies can be expressed as $\frac{C_4^3}{{}_{20}C_3}$.
Answer (2)
Calculate the probability of not choosing all comedies: 1 − ( 3 20 ) ( 3 4 ) = 1 − 1140 4 .
Calculate the probability of choosing three action movies: ( 3 20 ) ( 3 9 ) = 1140 84 .
Calculate the probability of not choosing all action movies: 1 − ( 3 20 ) ( 3 9 ) = 1 − 1140 84 .
Calculate the probability of choosing three comedies: ( 3 20 ) ( 3 4 ) = 1140 4 .
The probability that Christina will not choose all comedies can be expressed as I − 2 C b C b , where I = 1 , C b = 4 , and 2 C b = 20 C 3 = 1140 .
Explanation
Understand the problem We are given that Christina is choosing 3 movies from 9 action, 7 science fiction, and 4 comedies. The total number of movies is 9 + 7 + 4 = 20. We need to find which of the given probability statements is true.
Calculate probability of not choosing all comedies The probability that Christina will not choose all comedies is 1 − P ( all 3 are comedies ) .
We calculate P ( all 3 are comedies ) = ( 3 20 ) ( 3 4 ) = 1140 4 .
Therefore, the probability that Christina will not choose all comedies is 1 − 1140 4 = 1 − 285 1 = 285 284 ≈ 0.99649 .
Calculate probability of choosing three action movies The probability that Christina will choose three action movies is ( 3 20 ) ( 3 9 ) = 1140 84 = 95 7 ≈ 0.07368 .
Calculate probability of not choosing all action movies The probability that Christina will not choose all action movies is 1 − P ( all 3 are action ) .
We calculate P ( all 3 are action ) = ( 3 20 ) ( 3 9 ) = 1140 84 .
Therefore, the probability that Christina will not choose all action movies is 1 − 1140 84 = 1 − 95 7 = 95 88 ≈ 0.92632 .
Calculate probability of choosing three comedies The probability that Christina will choose three comedies is ( 3 20 ) ( 3 4 ) = 1140 4 = 285 1 ≈ 0.00351 .
Compare and conclude Now we compare the calculated probabilities with the given expressions.
The probability that Christina will not choose all comedies can be expressed as I − 2 C b C b .
We found this probability to be 1 − 1140 4 . If I = 1 and C b = 4 , then 2 C b = 20 C 3 = 1140 . So, 1 − 1140 4 matches the form 1 − 20 C 3 C b .
The probability that Christina will choose three action movies can be expressed as X C 3 C 1 .
We found this probability to be 1140 84 . If C 1 = 84 , then X C 3 = 20 C 3 = 1140 . So, 1140 84 matches the form X C 3 C 1 .
The probability that Christina will not choose all action movies can be expressed as 1 − 2 C 1 C 3 .
We found this probability to be 1 − 1140 84 . If C 3 = 84 , then 2 C 1 = 20 C 3 = 1140 . So, 1 − 1140 84 matches the form 1 − 20 C 3 C 3 .
The probability that Christina will choose three comedies can be expressed as C 3 1 .
We found this probability to be 1140 4 = 285 1 . If this is equal to C 3 1 , then C 3 = 285 . This does not match with any of the previous values. However, if we consider the expression to be ( 3 20 ) ( 3 4 ) = 1140 4 = 285 1 , then we can say that the probability of choosing three comedies can be expressed as ( 3 20 ) ( 3 4 ) .
Comparing the given options with the calculated probabilities, we can see that the first option is true.
Examples
This type of probability problem can be used in scenarios such as quality control, where you want to determine the probability of selecting a certain number of defective items from a batch. For example, if a factory produces 100 items, and 5 are defective, you can calculate the probability of selecting 3 items and having none of them be defective. This helps in assessing the quality of the production process and making informed decisions about whether to accept or reject a batch. The calculations involve combinations, as the order of selection does not matter. The formula for combinations is given by C ( n , k ) = k ! ( n − k )! n ! , where n is the total number of items and k is the number of items to be selected.
All the statements about Christina's choice of movies can be validated mathematically. Each option appropriately defines probabilities related to her selection of movies. However, option A is the most straightforward expression regarding the probability of not choosing all comedies.
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