Hey movie buffs! Let's dive into a fun probability problem involving Christina's vacation movie selections. She's got a mix of action, sci-fi, and comedy flicks, and we need to figure out the likelihood of her picking specific combinations. This is a classic example of how combinatorics and probability work together, and we're going to break it down step by step.
Understanding the Scenario
Christina has a total of nine action movies, seven science fiction movies, and four comedies. She's planning to randomly select three movies to bring on her vacation. The question we're tackling is: What's the probability that Christina will choose three comedies? To solve this, we'll need to use the concepts of combinations, which help us determine the number of ways to select items from a larger set without regard to order. Combinations are crucial when the order of selection doesn't matter, like in this case where the order in which Christina picks the movies doesn't change the final selection. We will explore the different possibilities and then calculate the probability. Understanding the total number of ways she can choose any three movies is our first step, followed by figuring out how many ways she can choose three comedies. By comparing these two numbers, we can determine the probability.
Calculating Total Possible Combinations
First, we need to calculate the total number of ways Christina can choose three movies out of her entire collection. She has 9 action movies + 7 science fiction movies + 4 comedies = 20 movies in total. We are choosing 3 movies out of 20, and the order doesn't matter, so we use combinations. The formula for combinations is:
nCr = n! / (r! * (n-r)!)
Where:
- n is the total number of items
- r is the number of items to choose
- ! denotes factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1)
In our case, n = 20 (total movies) and r = 3 (movies to choose). Plugging these values into the formula, we get:
20C3 = 20! / (3! * 17!)
Let's break this down. 20! means 20 * 19 * 18 * ... * 1, and 17! means 17 * 16 * 15 * ... * 1. When we divide 20! by 17!, we're left with 20 * 19 * 18 in the numerator. 3! is 3 * 2 * 1 = 6. So, the equation simplifies to:
20C3 = (20 * 19 * 18) / (3 * 2 * 1)
20C3 = (20 * 19 * 18) / 6
Now, we can simplify further by dividing 18 by 6, which gives us 3:
20C3 = 20 * 19 * 3
20C3 = 1140
So, there are 1140 different ways Christina can choose three movies from her collection. This is our denominator when we calculate the probability.
Calculating Combinations of Comedies
Next, we need to figure out how many ways Christina can choose three comedies out of the four she has. Again, we use the combinations formula, but this time n = 4 (total comedies) and r = 3 (comedies to choose):
4C3 = 4! / (3! * 1!)
4! is 4 * 3 * 2 * 1, 3! is 3 * 2 * 1, and 1! is just 1. So, the equation becomes:
4C3 = (4 * 3 * 2 * 1) / (3 * 2 * 1 * 1)
We can simplify this by canceling out 3 * 2 * 1 from both the numerator and the denominator:
4C3 = 4 / 1
4C3 = 4
There are 4 ways Christina can choose three comedies out of her four comedies. This is our numerator in the probability calculation.
Determining the Probability
Now that we know the total number of possible combinations (1140) and the number of ways to choose three comedies (4), we can calculate the probability. Probability is defined as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
In this case:
- Number of favorable outcomes = 4 (ways to choose three comedies)
- Total number of possible outcomes = 1140 (total ways to choose three movies)
So, the probability that Christina will choose three comedies is:
Probability = 4 / 1140
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
Probability = (4 / 4) / (1140 / 4)
Probability = 1 / 285
Therefore, the probability that Christina will choose three comedies is 1/285. This is a relatively small probability, which makes sense because comedies are a smaller portion of her overall movie collection.
Expressing the Probability
The probability can be expressed as a fraction (1/285), a decimal (approximately 0.0035), or a percentage (approximately 0.35%). Each of these representations gives us a different way to understand the likelihood of Christina choosing three comedies. The fraction 1/285 is the most precise form, while the decimal and percentage provide a more intuitive sense of the probability's magnitude.
Key Concepts Revisited
Let's recap the key concepts we used to solve this problem:
- Combinations: We used combinations because the order in which Christina chooses the movies doesn't matter. The formula nCr = n! / (r! * (n-r)!) is essential for calculating the number of ways to choose r items from a set of n items.
- Total Possible Outcomes: We calculated the total number of ways Christina could choose three movies from her entire collection of 20 movies. This served as the denominator in our probability calculation.
- Favorable Outcomes: We determined the number of ways Christina could choose three comedies from her four comedies. This served as the numerator in our probability calculation.
- Probability Formula: We applied the basic probability formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes).
These concepts are fundamental to solving a wide range of probability problems, especially those involving selections and arrangements.
Why This Matters
Understanding probability isn't just about solving math problems; it's a crucial skill for making informed decisions in everyday life. From assessing risks to understanding statistics, probability helps us make sense of the world around us. In this movie selection example, we see how probability can quantify the likelihood of specific events occurring, which can be applied to various real-world scenarios, such as games of chance, surveys, and even weather forecasting.
Additional Scenarios to Consider
To further explore this topic, let's consider some additional scenarios:
- What is the probability that Christina chooses one action movie, one science fiction movie, and one comedy?
- What is the probability that Christina chooses at least two action movies?
- If Christina chooses four movies instead of three, how does the probability of choosing all comedies change?
Each of these scenarios requires a slightly different approach, but the core concepts of combinations and probability remain the same. By working through these examples, we can deepen our understanding of these principles.
Final Thoughts
We've successfully navigated Christina's movie selection dilemma by applying the principles of combinations and probability. We calculated the total possible combinations, identified the favorable outcomes (choosing three comedies), and determined the probability. This problem demonstrates how mathematical concepts can be applied to real-life situations, and it highlights the importance of understanding probability in making informed decisions.
So, the next time you're faced with a selection problem, remember the power of combinations and probability! And who knows, maybe Christina will have a vacation filled with laughter, thanks to her choice of comedies (or maybe she'll mix it up with some action and sci-fi!).