Total Number of Ways to Select Three Movies

Oct 13, 2025 by miodeya.com 7 views

Introduction

Christina is preparing for a vacation and needs to choose three movies from a diverse collection of action, science fiction, and comedy films. In this scenario, we will explore the probability of Christina selecting three comedies from the given options. To begin, let's examine the total number of movies available and the specific categories.

Movie Collection

Christina has a total of 20 movies to choose from, consisting of:

9 action movies
7 science fiction movies
4 comedies

Calculating Probability

To determine the probability of Christina choosing three comedies, we need to consider the total number of ways she can select three movies from the entire collection and the number of ways she can select three comedies specifically.

Total Number of Ways to Select Three Movies

The total number of ways to select three movies from 20 can be calculated using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of movies (20), k is the number of movies to select (3), and ! denotes the factorial function.

C(20, 3) = 20! / (3!(20-3)!) = 20! / (3!17!) = (20 × 19 × 18) / (3 × 2 × 1) = 1140

Number of Ways to Select Three Comedies

The number of ways to select three comedies from the four available can be calculated using the combination formula:

C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 × 3 × 2 × 1) / (3 × 2 × 1) = 4

Probability of Choosing Three Comedies

The probability of Christina choosing three comedies can be expressed as the ratio of the number of ways to select three comedies to the total number of ways to select three movies:

P(3 comedies) = C(4, 3) / C(20, 3) = 4 / 1140 = 1 / 285

Conclusion

In conclusion, the probability that Christina will choose three comedies can be expressed as 1/285. This result highlights the low likelihood of selecting three comedies from the given collection of movies.

Discussion

This problem involves understanding the concept of probability and applying it to a real-world scenario. The combination formula is used to calculate the total number of ways to select three movies and the number of ways to select three comedies. The probability is then determined by dividing the number of ways to select three comedies by the total number of ways to select three movies.

Key Takeaways

The probability of an event is a measure of the likelihood of that event occurring.
The combination formula is used to calculate the number of ways to select items from a set.
The probability of an event can be expressed as a ratio of the number of favorable outcomes to the total number of possible outcomes.

Further Exploration

This problem can be extended to explore other scenarios, such as:

What is the probability of Christina choosing three action movies?
What is the probability of Christina choosing three science fiction movies?
What is the probability of Christina choosing three movies from a specific genre?

Q: What is the total number of movies available for Christina to choose from?

A: Christina has a total of 20 movies to choose from, consisting of 9 action movies, 7 science fiction movies, and 4 comedies.

Q: How many ways can Christina select three movies from the entire collection?

A: The total number of ways to select three movies from 20 can be calculated using the combination formula: C(20, 3) = 1140.

Q: How many ways can Christina select three comedies from the four available?

A: The number of ways to select three comedies from the four available can be calculated using the combination formula: C(4, 3) = 4.

Q: What is the probability of Christina choosing three comedies?

A: The probability of Christina choosing three comedies can be expressed as 1/285, which is the ratio of the number of ways to select three comedies to the total number of ways to select three movies.

Q: What is the significance of the combination formula in this problem?

A: The combination formula is used to calculate the total number of ways to select three movies and the number of ways to select three comedies. This formula is essential in determining the probability of Christina choosing three comedies.

Q: Can the probability of Christina choosing three comedies be expressed as a percentage?

A: Yes, the probability of Christina choosing three comedies can be expressed as a percentage by dividing the probability by 1 and multiplying by 100: (1/285) × 100 = 0.35%.

Q: What is the implication of the low probability of Christina choosing three comedies?

A: The low probability of Christina choosing three comedies suggests that it is unlikely for her to select three comedies from the given collection of movies.

Q: Can the problem be extended to explore other scenarios?

A: Yes, the problem can be extended to explore other scenarios, such as calculating the probability of Christina choosing three action movies, three science fiction movies, or three movies from a specific genre.

Q: What is the relevance of this problem in real-world situations?

A: This problem is relevant in real-world situations where probability is used to make informed decisions. For example, in marketing, probability can be used to determine the likelihood of a customer purchasing a product based on their past behavior.

Q: Can the combination formula be used to solve other problems involving probability?

A: Yes, the combination formula can be used to solve other problems involving probability, such as calculating the probability of selecting a certain number of items from a set.

Q: What is the importance of understanding probability in real-world situations?

A: Understanding probability is essential in real-world situations where decisions need to be made based on uncertain outcomes. Probability can be used to make informed decisions and reduce the risk of uncertainty.