Question:

In employment discrimination cases, courts have held that a firm is guilty of discrimination when

the percentage of Black people among its employees is lower than the percentage of Black people in the surrounding geographical region, provided the difference is "statistically significant" (i.e., p-value less than 5%) by the z-test.

1. Consider a metropolitan area in which 10% of the people are Black, and 50 firms hire employees independently by a process which, as far as race is concerned, is equivalent to simple random sampling (note this can be true without hiring decisions being made completely at random). That is, the probability of each new hire being Black is independently 10%. What is the chance that at least one of the 50 firms will be found guilty of discrimination? You may assume that each firm hires enough people that a z-test is appropriate.
2. What is the expected number of firms that will be found guilty?
3. Now to a meta level: One might be skeptical of our assumption that, with respect to race, these 50 employers all hire by simple random sampling. Let us take this assumption as a null hypothesis and test it. Suppose that 7 out of the 50 firms are found guilty of discrimination. Should we reject the null hypothesis? Back up your answer by performing an appropriate test of significance and computing the p-value.

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