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Scholarships are an essential part of many students’ journeys through higher education. They provide financial support and can significantly reduce the burden of tuition fees and other educational expenses. However, the process of receiving a scholarship can be quite competitive, and understanding the probabilities involved can help students make informed decisions and manage their expectations effectively.

In this article, we will explore the concept of the probability of scholarship awards. Specifically, we will discuss how probabilities are used to assess the likelihood of receiving a scholarship from different agencies and how these probabilities can be combined to determine the overall chance of being awarded at least one scholarship.

## The Scenario

Let’s begin by setting up a scenario. A student is applying to two different agencies for scholarships. Based on the student’s academic record, the probability that the student will be awarded a scholarship from agency A is 0.55, and the probability that the student will be awarded a scholarship from agency B is 0.40. Furthermore, if the student is awarded a scholarship from agency A, the probability that the student will be awarded a scholarship from agency B is 0.60. What is the probability that the student will be awarded at least one of the two scholarships? To make our discussion more concrete, we will assign probabilities to various events in this scenario.

## Probability of Receiving a Scholarship from Agency A

First, let’s consider the probability that the student will be awarded a scholarship from Agency A. We’ll denote this probability as P(A), and in our scenario, let’s assume P(A) = 0.55. This means that there is a 55% chance that the student will receive a scholarship from Agency A based on their academic record.

## Probability of Receiving a Scholarship from Agency B

Next, we’ll examine the probability of receiving a scholarship from Agency B, denoted as P(B). In our scenario, we’ll assume P(B) = 0.40, indicating a 40% chance of receiving a scholarship from Agency B based on the same academic record.

## Conditional Probability

Now, things get a bit more interesting when we consider conditional probability. Conditional probability involves the probability of an event happening given that another event has already occurred. In our scenario, we want to find the probability that the student will be awarded a scholarship from Agency B, given that they have already received a scholarship from Agency A.

This conditional probability is denoted as P(B|A), where “B|A” means “B given A.” We’ll assume P(B|A) = 0.60, indicating a 60% chance of receiving a scholarship from Agency B if the student has already received one from Agency A.

## Finding the Probability of Receiving at Least One Scholarship

Now that we have these probabilities established, our primary question is: What is the probability that the student will be awarded at least one of the two scholarships? To answer this question, we can use the concept of the complement rule.

The complement rule states that the probability of an event happening (in this case, receiving at least one scholarship) is equal to 1 minus the probability of the event not happening (receiving no scholarships). So, we can calculate the probability of receiving no scholarships from both agencies and subtract it from 1 to find the probability of receiving at least one scholarship.

## Probability of Receiving No Scholarships

To find the probability of receiving no scholarships, we need to consider two mutually exclusive events:

- Not receiving a scholarship from Agency A (denoted as P(not A)).
- Not receiving a scholarship from Agency B given that a scholarship was not received from Agency A (denoted as P(not B|not A)).

Let’s calculate these probabilities step by step.

## Probability of Not Receiving a Scholarship from Agency A

P(not A) is the complement of P(A), meaning it’s the probability of not receiving a scholarship from Agency A. In our scenario, P(A) = 0.55, so P(not A) = 1 – P(A) = 1 – 0.55 = 0.45.

## Probability of Not Receiving a Scholarship from Agency B Given Not Receiving from Agency A

Now, we need to find P(not B|not A), which is the probability of not receiving a scholarship from Agency B given that a scholarship was not received from Agency A.

Since these events are independent (the outcome of one doesn’t affect the other), we can calculate P(not B|not A) simply as P(not B), which is the probability of not receiving a scholarship from Agency B.

In our scenario, P(B) = 0.40, so P(not B) = 1 – P(B) = 1 – 0.40 = 0.60.

Now, we have the probabilities we need to find the probability of receiving no scholarships:

P(not A) * P(not B) = 0.45 * 0.60 = 0.27.

This means there is a 27% chance that the student will not receive a scholarship from either Agency A or Agency B.

## Finding the Probability of Receiving at Least One Scholarship

Finally, we can use the complement rule to find the probability of receiving at least one scholarship:

P(at least one scholarship) = 1 – P(no scholarships) = 1 – 0.27 = 0.73.

Therefore, there is a 73% chance that the student will be awarded at least one of the two scholarships when applying to both Agency A and Agency B.

## Conclusion

Understanding the probabilities of scholarship awards can help students make informed decisions about their applications and manage their expectations. In our scenario, we explored the probabilities of receiving scholarships from two agencies and used conditional probability and the complement rule to calculate the probability of receiving at least one scholarship.

Keep in mind that these probabilities are hypothetical and may vary in real-life situations. Nevertheless, having a grasp of these concepts can empower students to make informed choices and approach scholarship applications with confidence.

In conclusion, while scholarships are competitive, a well-informed approach can increase your chances of receiving the financial support you need to pursue your educational goals.