To solve these problems, we need to apply conditional probability formulas and the given data on male and female students who received business degrees. Since you haven't provided the actual data, I’ll explain how to calculate the probabilities based on hypothetical data. If you have the actual numbers, you can substitute them into the formulas provided below.

Let's define the following variables:

• $M$: Total number of male students who received business degrees.
• $F$: Total number of female students who received business degrees.
• $T$: Total number of students (both male and female) who received business degrees, i.e., $T = M + F$.

Part (a) Probability that a student is male, given that the student received a business degree

The conditional probability formula is: $P(\text{Male} \mid \text{Business Degree}) = \frac{P(\text{Male and Business Degree})}{P(\text{Business Degree})}$ Since we are only dealing with students who received a business degree, we simplify this as: $P(\text{Male} \mid \text{Business Degree}) = \frac{\text{Number of Male Business Graduates}}{\text{Total Number of Business Graduates}} = \frac{M}{T}$ If you have the numbers for male students ($M$) and total students ($T$), you can calculate this probability.

Part (b) Probability that a student received a business degree, given that the student is female

Here, we use the conditional probability formula: $P(\text{Business Degree} \mid \text{Female}) = \frac{P(\text{Female and Business Degree})}{P(\text{Female})}$ This simplifies to: $P(\text{Business Degree} \mid \text{Female}) = \frac{\text{Number of Female Business Graduates}}{\text{Total Number of Female Students}} = \frac{F}{F_{\text{total}}}$ where $F_{\text{total}}$ is the total number of female students (in any field, not just business). Again, substitute the data to compute this probability.

Example with Hypothetical Numbers:

Assume:

• $M = 6000$ (number of males who received business degrees),

• $F = 8000$ (number of females who received business degrees),

• $T = M + F = 14000$ (total number of business graduates).

• Total number of female students in any field, $F_{\text{total}} = 25000$.

(a) Probability that a randomly selected student is male, given that they received a business degree:

$P(\text{Male} \mid \text{Business Degree}) = \frac{6000}{14000} = 0.429$ So, the probability is $0.429$.

(b) Probability that a randomly selected student received a business degree, given that the student is female:

$P(\text{Business Degree} \mid \text{Female}) = \frac{8000}{25000} = 0.32$ So, the probability is $0.320$.

Once you provide the actual numbers, I can help refine the calculations.

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Would you like to provide the data for a more precise answer? Let me know if you have any questions or need further clarification.

Here are 5 related questions to explore further:

1. What is the overall probability that a randomly selected student received a business degree?
2. What is the probability that a randomly selected student is female, given that they received a business degree?
3. How would the probabilities change if the number of male graduates increased by 10%?
4. What is the probability that a randomly selected student is female, considering all students?
5. How would the calculations differ if we also included students who received degrees in other fields?

Tip: When dealing with conditional probabilities, always make sure you clearly define the total group relevant to your condition.