Consider an 8-ball lottery game. In total there are 8 balls numbered 1 through to 8 inclusive. 5 balls are drawn

(chosen randomly), one at a time, without replacement (so that a ball cannot be chosen more than once). To

win the grand prize, a lottery player must have the same numbers selected as those that are drawn. Order of

the numbers is not important so that if a lottery player has chosen the combination

1, 2, 3, 4, 5

and, in order, the numbers

4, 2, 3, 5, 1

are drawn, then the lottery player will win the grand prize (to be shared with other grand prize winners). You

can assume that each ball has exactly the same chance of being drawn as each of the others.

(a) Consider a population of size N = 8. How many dierent random samples of size n = 5 are possible from

a population of N = 8?

(b) Suppose that you choose the numbers 1, 2, 3, 4 and 5 ahead of the next lottery draw. As a fraction, what is

the exact probability that you will win the grand prize in the lottery in the next draw with these numbers?

(c) Continuing on part (b) and again as a fraction, what is the exact probability that you will not win the

lottery in the next draw with these numbers? Show workings.

(d) This question is tougher and you need to think carefully about the answer. Recall that the order of the

numbers chosen is not important and that each number can only be chosen once. In total, how many

combinations are there available that include the numbers 1 and 2 but not the numbers 3, 4 and 5?

Explain and show workings. (Hint: The available combinations must include the numbers 1 and 2 and

they may also include the numbers 3, 4 or 5, but they must not include the numbers 3, 4 and 5 in one

combination at the same time.)