Question

There are 30 students in your class. Your science teacher chooses 5 students at random to complete a group project. Find the probability that you and your 2 best friends in the science class are chosen to work in the group. Explain how you found your answer.

Short answer

The answer can be found by first calculating the total possible groups of 5 students using the combination formula, then calculating in how many of these groups you and your 2 best friends are always included. The probability is then found from the ratio of these two numbers.

Step-by-step solution

Calculate Total Possible Groups

Firstly, calculate the total possible ways the teacher can form a group of 5 students from 30. This is represented by the combination \(C(30,5)\), which can be calculated using the formula \(\frac{n!}{r!(n-r)!}\), where n = 30 (total students), r = 5 (students in group), and '!' means factorial, which is the product of an integer and all integers below it. Set \(n = 30\) and \(r = 5\) in the formula to get the total combinations.

Calculate Favored Outcomes

Next, find in how many of these groups you and your 2 best friends are always included. If all 3 of you are in the group, the other 2 members can be any of the remaining 27 students. This is represented by the combination \(C(27,2)\).

Calculate the Probability

The probability is given by the ratio of favored outcomes to total outcomes. It can be calculated using the formula \(\frac{favored}{total}\). Using the results from step 1 and step 2, substitute these values into the probability formula.

Key concepts

Understanding Combinations

When solving problems that involve selecting items from a group, such as choosing students for a project from a class, we often use the concept of combinations. Combinations refer to the number of ways to select items where the order does not matter. For example, if you're choosing 5 students out of a class of 30 for a group project, it is a combination because the order in which you pick the students does not affect the group composition.

In solving our problem, we're not just looking at simple combinations but also at a conditional situation where you and your two best friends must be in the group. The combination formula \(\frac{n!}{r!(n-r)!}\) is essential where \(n\) represents the total number of items to choose from, and \(r\) is the number of items to be chosen. By using this formula, we can calculate all possible groups without concerning ourselves with the order in which they're chosen, which is pivotal in probability calculations related to random selections.

Factorial Notation in Calculations

Factorial notation is a mathematical expression used frequently in probability and combinations. The factorial of a positive integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). In simpler terms, if \(n=5\), then \(5! = 5 \times 4 \times 3 \times 2 \times 1\).

Understanding factorial notation is critical when working with combinations because the formula for calculating combinations includes factorials. They represent the total number of ways to arrange a certain number of items and are foundational in determining the number of possible outcomes in many probability exercises. Factorials grow at an extremely fast rate, so the numbers can become very large, very quickly, which is why it's important to simplify the combination formula as much as possible before calculating.

Probability Calculation and Ratio

In our science class project example, the calculation of probability is the final step. Probability measures how likely it is for an event to occur and is calculated as the number of favorable outcomes divided by the total number of possible outcomes. This can be expressed as a fraction, a decimal, or a percentage.

For our exercise, after finding the total possible groups (\(C(30,5)\)) and the groups where you and your two friends are included (\(C(27,2)\)), the probability is the ratio of these two. In essence, calculating probability is about quantifying the chance of a specific condition or set of conditions being met within the context of a larger set of possibilities. It's a fundamental concept in statistics and carries immense practical importance across various fields – not just in classroom settings but in real-world scenarios like predicting weather events, understanding market trends, and even in sports and gaming strategies.