A class in advanced physics is composed of 10
juniors, 30 seniors, and 10 graduate students. The “nal
grades show that 3 of the juniors, 10 of the seniors, and
5 of the graduate students received anA for the course.
If a student is chosen at random from this class and is
found to have earned anA, what is the probability that
he or she is a senior?

To solve this problem, we'll use the concept of conditional probability. The question asks for the probability that a student is a senior given that the student received an A. 

We will use Bayes' theorem for conditional probability:

\[ P(\text{Senior | A}) = \frac{P(\text{A | Senior}) \times P(\text{Senior})}{P(\text{A})} \]

First, let's define the given information:
- Number of juniors = 10
- Number of seniors = 30
- Number of graduate students = 10
- Number of juniors who received an A = 3
- Number of seniors who received an A = 10
- Number of graduate students who received an A = 5

The total number of students in the class:
\[ 10 + 30 + 10 = 50 \]

The total number of students who received an A:
\[ 3 + 10 + 5 = 18 \]

Next, we calculate each part of Bayes' theorem:

1. \( P(\text{A | Senior}) \) - The probability that a senior received an A:
\[ P(\text{A | Senior}) = \frac{\text{Number of seniors who received an A}}{\text{Total number of seniors}} = \frac{10}{30} = \frac{1}{3} \]

2. \( P(\text{Senior}) \) - The probability of selecting a senior from the class:
\[ P(\text{Senior}) = \frac{\text{Total number of seniors}}{\text{Total number of students}} = \frac{30}{50} = \frac{3}{5} \]

3. \( P(\text{A}) \) - The overall probability of a student receiving an A:
\[ P(\text{A}) = \frac{\text{Total number of students who received an A}}{\text{Total number of students}} = \frac{18}{50} = \frac{9}{25} \]

Now, we apply these values to Bayes' theorem:

\[ P(\text{Senior | A}) = \frac{P(\text{A | Senior}) \times P(\text{Senior})}{P(\text{A})} \]
\[ P(\text{Senior | A}) = \frac{\left(\frac{1}{3}\right) \times \left(\frac{3}{5}\right)}{\left(\frac{9}{25}\right)} \]

Simplify the fraction:

\[ P(\text{Senior | A}) = \frac{\left(\frac{1}{3} \times \frac{3}{5}\right)}{\frac{9}{25}} = \frac{\left(\frac{1}{5}\right)}{\frac{9}{25}} = \frac{1}{5} \times \frac{25}{9} = \frac{25}{45} = \frac{5}{9} \]

Therefore, the probability that a student chosen at random who received an A is a senior is \( \frac{5}{9} \).

Would you like more details or have any questions about this solution?

Here are some related questions for further practice:

1. What is the probability that a randomly selected student is a junior?
2. What is the probability that a randomly selected student received an A?
3. What is the probability that a student is a junior given that the student received an A?
4. What is the probability that a student is a graduate student given that the student received an A?
5. What is the probability that a student received an A given that they are a junior?
6. What is the probability that a student received an A given that they are a graduate student?
7. If a student is known to be a senior, what is the probability that they received an A?
8. If a student is known to be a graduate student, what is the probability that they received an A?

**Tip:** When dealing with conditional probabilities, it's important to clearly define each event and use the relationships between them to set up your calculations correctly.

Calculate the probability that a randomly chosen student who earned an A in an advanced physics class is a senior using Bayes' theorem and conditional probability concepts. Includes step-by-step solution and relevant mathematical formulas.

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