Let's solve the probability questions based on the given data from the table. First, I'll extract the values from the table and solve each question one by one.

Table Summary:

• Freshman:
  • A: 107
  • B: 128
  • C: 100
• Sophomore:
  • A: 149
  • B: 117
  • C: 138

Total Students:

• Freshmen: $107 + 128 + 100 = 335$
• Sophomores: $149 + 117 + 138 = 404$
• Total Students = $335 + 404 = 739$

Now let's solve each part:

a) Probability that the student was a sophomore:

The probability of choosing a sophomore is the ratio of sophomores to the total number of students.

$P(\text{sophomore}) = \frac{404}{739} \approx 0.5467$

b) Probability that the student was a freshman or received a C:

We need to find the probability that the student was either a freshman or received a grade of C.

• Total freshmen: $335$
• Total students who received a C: $100 + 138 = 238$
• Freshmen who received a C: $100$ (this needs to be subtracted to avoid double counting).

So, the probability is:

$P(\text{freshman or C}) = \frac{335 + 238 - 100}{739} = \frac{473}{739} \approx 0.6400$

c) Probability that the student was a sophomore, given they got a C:

This is a conditional probability. We need to find $P(\text{sophomore} | \text{C})$, which is given by:

$P(\text{sophomore} | \text{C}) = \frac{P(\text{sophomore and C})}{P(\text{C})}$

• Students who got a C: $238$
• Sophomores who got a C: $138$

$P(\text{sophomore} | \text{C}) = \frac{138}{238} \approx 0.5798$

d) Probability that the student got a B, given they are a freshman:

This is another conditional probability. We need to find $P(\text{B} | \text{freshman})$, which is given by:

$P(\text{B} | \text{freshman}) = \frac{P(\text{freshman and B})}{P(\text{freshman})}$

• Freshmen who got a B: $128$
• Total freshmen: $335$

$P(\text{B} | \text{freshman}) = \frac{128}{335} \approx 0.3821$

Summary of Answers:

a) $0.5467$

b) $0.6400$

c) $0.5798$

d) $0.3821$

Let me know if you would like more details or explanations about any of the steps! Here are some related questions to expand on this topic:

1. How do we calculate the total probability for multiple groups?
2. What is the difference between joint probability and conditional probability?
3. How would the results change if there was an additional grade, say D?
4. Can we apply these probabilities to real-life scenarios involving categories of students?
5. How does the conditional probability help in making predictions?

Tip:

Conditional probabilities help in narrowing down the possible outcomes based on additional information, making them very useful in statistical inference.