A company wants to create a new four-member committee that reviews annual salary increases for all...

A company wants to create a new four-member committee that reviews annual salary increases for all employees. The committee members have to be chosen from 12 existing managers in the company, 8 of these managers are men. The selection committee states that all twelve managers have equal chances of being picked since they are all equally qualified. For concerns of fairness, the salary-review committee should have at least one women member. What is the probability that in a randomly selected committee, where fairness concerns are not exercised, at least one member will be a woman? (If deliberate fairness concerns were exercised, the selection committee would make sure to pick a woman and the probability of a woman being in the committee is 1.) Ultimately we see that the committee is all female. The selection committee insists this was a random occurrence and not a deliberate attempt to only pick women. (Again, if it were a deliberate attempt, then the probability of an all-woman committee would be 1.) Do you believe them? Why or why not?

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