1. Consider a 9ball lottery game. In total there are 9 balls numbered 1 through to 9 inclusive. 6 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. The order of the numbers is not important, so that if for example a lottery player has chosen the combination 1, 2, 3, 4, 5, 6 and, in order, the numbers 6,4,2,3,5, 1 are drawn, then the lottery player will still 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 = 9. How many dierent random samples of size n = 6 are possible from a population of N = 9? (b) Suppose that you choose the numbers 1, 2, 3, 4, 5 and 6 ahead of the next lottery draw. i. 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? ii. What is the exact probability that you will not win the lottery in the next draw with these numbers? Make sure to show all your workings. (c) Recall that the order of the numbers chosen in the 9ball lottery game 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, 5 and 6? Make sure to show all your workings. Hint: This question is tougher and you will need to think: carefully about the answer. The auailable com- binations must include the numbers 1 andr 2 and they may also include the numbers 3, 4, 5, or 6, but must not include the numbers 3, 4, 5 and 6 in one combination at the same time