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

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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.