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When you're randomly selecting a group of items or individuals from a larger population, you'll often need to figure out how many different ways you can make those selections. This is where the combination formula comes in handy. The combination formula helps determine how many ways you can select a certain number of items from a larger set without worrying about the order of selection. In mathematics, this is represented as \( \binom{n}{r} \), where \( n \) is the total number of items in the set, and \( r \) is the number of items you want to select.

For example, in the context of our problem, where we are selecting 2 students from a total of 90, the formula is used as follows: \( \binom{90}{2} \). Plugging in the numbers, the calculation is: \( \frac{90 \times 89}{2} = 4005 \). This tells us there are 4,005 possible ways to choose 2 students from 90.

This formula is useful in many probability problems as it allows us to discern the different possible outcomes without considering the sequence of selection.

Probability calculation in problems like this involves determining the likelihood of a specific event occurring out of all possible events. To calculate probability, you divide the number of favorable outcomes by the total number of possible outcomes. It's portrayed as a ratio, a fraction, or a percentage.

In this scenario, there are two main probabilities to calculate:

• Both students attending college: First, calculate how many ways you can choose 2 students from the 50 who plan to attend college. This is given by \( \binom{50}{2} = \frac{50 \times 49}{2} = 1225 \). Thus, the probability is \( \frac{1225}{4005} \approx 0.3059 \), or roughly 30.59%.
• Exactly one student attending college: Calculate the number of ways to pick one student from each group (those planning and not planning to attend), which is \( 50 \times 40 = 2000 \). This results in a probability of \( \frac{2000}{4005} \approx 0.4994 \), or approximately 49.94%.

Breaking down probability in this manner not only helps in solving specific problems but builds a foundation for understanding complex situations.

In our exercise, we observe vital statistics about college attendance which influences our probability calculations. Out of 90 graduating students, 50 plan to attend college while 40 do not. These statistics provide a straightforward way to separate our sample into two distinct groups.

College attendance statistics like these are more than just numbers; they reflect larger trends and decisions made by students. They help us understand the choices individuals make and how these might influence other scenarios, like job offers or location preferences after graduation.

Understanding and analyzing these statistics are crucial not only in probability problems but also in making informed predictions and decisions in education and policy making.

Random selection plays a crucial role in probability and statistics, especially in educational settings like our problem scenario. When we say you randomly select students, it means each student has an equal chance of being chosen, ensuring that the selection process is fair and unbiased.

The core principle of random selection is its unpredictability and evenness; every individual has the same opportunity to be selected irrespective of characteristics like choices or grades. In our example, it signifies that any of the 90 students could be selected to carry flags without any predetermined conditions or preferences.

Random selection is a cornerstone of probability studies, aiding in simulations, modeling, and understanding real-world statistics. It reinforces the integrity of statistical findings by eliminating potential biases in sample selection.