Stanford has 550 Law students who will soon become lawyers. A large law firm has decided to hire 10 of these students. 306 of those students are female while 244 were male. Of the 10 law students they chose to hire, 3 of them were women.
1. What is the sample space for the situation above
2. How could we design a simulation to test the probability of this happening out of 1000 trials?
3. How many women would we expect to be hired assuming all Stanford graduates are equally qualified?
4. Create a probability distribution that shows the probability of hiring between 0 and 10 women.
5. What is the theoretical probability that a law school would hire 3 or fewer women from Stanford's graduating class out of their 10 hires? Use the probability distribution to answer this question.
6.Based on your results from your work earlier, is there evidence that the law firm had sexist hiring practices?

The sample space represents all possible outcomes of the situation. In this case, the sample space consists of all possible combinations of 10 law students chosen from the total pool of 550 Stanford law students.

2. **Simulation design:**

To simulate this situation, we could randomly select 10 students from the pool of 550 Stanford law students multiple times (e.g., 1000 trials). Each time, we would record the number of women among the 10 students chosen. By repeating this process multiple times, we can estimate the probability of hiring 3 or fewer women out of 10 hires.

3. **Expected number of women hired:**

Assuming all Stanford graduates are equally qualified and gender does not play a role in hiring decisions, we would expect the proportion of women hired to be equal to the proportion of women in the total pool of law students.

Expected number of women hired = Total number of hires * Proportion of women in the pool = 10 * (306/550) ≈ 5.33

4. **Probability distribution:**

Number of Women Hired | Probability

----------------------|------------

0 | Calculated using combinations

1 | Calculated using combinations

2 | Calculated using combinations

3 | Calculated using combinations

4 | Calculated using combinations

5 | Calculated using combinations

6 | Calculated using combinations

7 | Calculated using combinations

8 | Calculated using combinations

9 | Calculated using combinations

10 | Calculated using combinations

5. **Theoretical probability:**

To find the probability of hiring 3 or fewer women out of 10 hires, we would sum the probabilities of hiring 0, 1, 2, and 3 women from the probability distribution calculated in question 4.

6. **Evidence of sexist hiring practices:**

Based on the theoretical probability and the simulation results, if the actual hiring outcome significantly deviates from the expected outcome without a reasonable explanation (such as bias-free hiring), it could be suggestive of sexist hiring practices. However, further analysis and investigation would be needed to draw definitive conclusions.