**An exam was given to a group of freshman and sophomore students. The results are summarized below:**



**A**

**B**

**C**

**Freshman**

124

165

110

**Sophomore**

156

191

180


**If one student is chosen at random from those who took the exam, find the probability that:**
*Round all answers to 4 decimal places as needed*

**a)The student was a sophomore.**


**b)The student was a freshman or received a C.**


**c) The student was a sophomore, given they got a C.**


**d) The student got a B, given they are a freshman.**

To solve these probability questions, we will first calculate the total number of students and then break down the problem step by step using basic probability formulas.

### Step 1: Calculate the Total Number of Students
The total number of students can be obtained by summing all the entries in the table:

- Freshmen: \( 124 + 165 + 110 = 399 \)
- Sophomores: \( 156 + 191 + 180 = 527 \)

Thus, the total number of students is:
\[
399 + 527 = 926
\]

### Step 2: Solve the Probability Questions

#### a) The student was a sophomore.
The probability of selecting a sophomore is the ratio of sophomores to the total number of students:
\[
P(\text{Sophomore}) = \frac{\text{Total Sophomores}}{\text{Total Students}} = \frac{527}{926}
\]
Now, let's calculate this:
\[
P(\text{Sophomore}) = 0.5693
\]

#### b) The student was a freshman or received a C.
To find this probability, we need to find the total number of freshmen and the total number of students who received a C (from both freshmen and sophomores). Then we'll use the inclusion-exclusion principle to avoid double-counting the freshmen who also received a C.

- Freshmen: \( 399 \)
- Students who received a C: \( 110 + 180 = 290 \)
- Freshmen who received a C (double-counted): \( 110 \)

Thus, the probability is:
\[
P(\text{Freshman or C}) = \frac{(\text{Freshmen} + \text{C Students} - \text{Freshmen who got a C})}{\text{Total Students}} = \frac{399 + 290 - 110}{926} = \frac{579}{926}
\]
Now, calculate this:
\[
P(\text{Freshman or C}) = 0.6253
\]

#### c) The student was a sophomore, given they got a C.
This is a conditional probability, given by:
\[
P(\text{Sophomore | C}) = \frac{P(\text{Sophomore and C})}{P(\text{C})}
\]
Where:
- \( P(\text{Sophomore and C}) = \frac{180}{926} \)
- \( P(\text{C}) = \frac{290}{926} \)

Thus:
\[
P(\text{Sophomore | C}) = \frac{180/926}{290/926} = \frac{180}{290}
\]
Now, calculate this:
\[
P(\text{Sophomore | C}) = 0.6207
\]

#### d) The student got a B, given they are a freshman.
This is another conditional probability:
\[
P(\text{B | Freshman}) = \frac{P(\text{Freshman and B})}{P(\text{Freshman})}
\]
Where:
- \( P(\text{Freshman and B}) = \frac{165}{926} \)
- \( P(\text{Freshman}) = \frac{399}{926} \)

Thus:
\[
P(\text{B | Freshman}) = \frac{165/926}{399/926} = \frac{165}{399}
\]
Now, calculate this:
\[
P(\text{B | Freshman}) = 0.4135
\]

### Final Answers
a) The probability that the student was a sophomore: **0.5693**

b) The probability that the student was a freshman or received a C: **0.6253**

c) The probability that the student was a sophomore, given they got a C: **0.6207**

d) The probability that the student got a B, given they are a freshman: **0.4135**

---

Do you want more details or have any questions on this problem?

Here are 5 related questions:
1. What is the probability that the student was a freshman, given they received an A?
2. What is the probability that the student received a B?
3. What is the probability that the student received an A, given they are a sophomore?
4. What is the probability that the student is a freshman or sophomore?
5. What is the probability that a student who got an A is a freshman?

**Tip:** Conditional probability depends on focusing on a specific group as the denominator instead of the entire sample space.

An exam was given to a group of freshman and sophomore students. The results are summarized below:

A B C
Freshman 124 165 110
Sophomore 156 191 180

If one student is chosen at random from those who took the exam, find the probability that:
1. The student was a sophomore.
2. The student was a freshman or received a C.
3. The student was a sophomore, given they got a C.
4. The student got a B, given they are a freshman.

Find the probability of a randomly selected student being a sophomore, or receiving certain grades like A, B, or C in a class of freshmen and sophomores. Learn how to calculate using the inclusion-exclusion principle and conditional probability.

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