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The CIA wishes to hire 2 security experts to add to its work force.  In response to the CIA's job advertisements, they receive 5 applications. The applicants, which we will number 1 through 5, vary in their security competencies, with 1 being the most competent, and 5 being the least.  The CIA does not know any of the applicants' competency levels.

We define two events, A and B such that:

A: The CIA selects the most competent applicant, and one of the two least competent applicants.  (i.e., applicant 1 plus one of applicants 4 and 5).

B: The CIA selects at least one of the two most competent applicants. (i.e., applicant 1 plus any other candidate, or applicant 2 plus any other candidate)

Find the probabilities of each of these events occurring.

Here are the steps to accomplish this:

1. The process ("experiment") involves randomly selecting a pair of applicants out of the total 5 applicants.  Denote the selection of a pair of applicants by {i, j}.  For example, the selection of applicants 2 and 4 is denoted by {2, 4}.  

   Note that ordering is not important, so {4, 2} is equivalent to {2, 4}.

2. Write down all the possible pair selections ("events"):

   E₁: {1, 2},    E₂: {1, 3},  ...  etc.

   How many possible selections do you have in total?

3. Because we are assuming a random selection of the 2 out of the 5 applicants, each of the pairs you wrote down in the previous step has an equal chance for selection.  So, assign each of the above pairs an equal probability. 

   If you add up all these probabilities, what value should you arrive at? (Think in terms of what constitutes certainty: the CIA will choose one of these pairs.)

   Given how many pairs you have in total, and that each pair has the same probability as any other, what is the probability of a given pair being selected?

4. Given the events (sample points) E_i you arrived at in step b., how many of them satisfy the event A ?

   So what is the probability, P(A) of event A occurring?

5. Given the events E_i, how many satisfy the event B?

   So, what is the probability, P(B), of event B occurring?