The possible outcomes are: AB, AC, AD, BC, BD, CD.

[There are a few explanations for each answer - hopefully at least one of them makes sense!]

#### Part (a)

Explanation 1: The probability is `3/6=1/2` since when we choose A, we must choose one of the remaining 3 directors to go with A. There are `C_2^4=6` possible combinations.

Explanation 2: Probability that A is selected is `{C_1^1 times C_1^3}/{C_2^4} = 3/6 = 1/2`

[Choose A (`C_1^1`), and then choose one from the 3 remaining directors (`C_1^3`), divided by the number of possible outcomes: `C_2^4`.]

#### Part (b)

Explanation 1: The probability of getting A or B first is `2/4=1/2`.

Now to consider the probability of selecting A or B as the second director. In this case, the first director has to be C or D with probability `2/4` (2 particular directors out of 4 possible).

Then the probability of the second being A or B is `2/3` (2 particular directors out of the remaining 3 directors).

We need to multiply the two probabilities.

So the probability of getting A or B for the second director is `2/4 xx 2/3 = 1/3`

The total is: `1/2 + 1/3 = 5/6`

Explanation 2: Probability that A or B is selected is

`frac{C_1^1 times C_1^3 + C_1^1 times C_1^2}{C_2^4}` `=frac{3+2}{6}` `=5/6`

[Choose A as above, then choose B from the remaining 2 directors in a similar way.]

Explanation 3: If A or B is chosen, then we cannot have the case C and D is chosen. So the probability of A or B is given by:

`P("A or B") = 1-P("C and D")` `=1-1/6` `=5/6`

#### Part (c)

Probability that A is not selected is `1-1/2=1/2`

### Extension

Consider the case if we are choosing 2 directors from 5. The probabilities would now be:

(a) Probability that A is selected is

`frac{C_1^1 times C_1^4}{C_2^5}=4/10=2/5`

[Choose A (`C_1^1`), and then choose one from the 3 remaining directors (`C_1^4`), divided by the number of possible outcomes: `C_2^5`.]

(b) Probability that A or B is selected is

`frac{C_1^1 times C_1^4 + C_1^1 times C_1^3}{C_2^5}` `=frac{4+3}{10}` `=7/10`

[Choose A as above and then choose B from the remaining 3].

(c) Probability that A is not selected is `1-2/5=3/5`.