Roulette is a game that can be found at many casinos around the world. There can be slight variations in the game depending on country and we will be using American Roulette for this activity. A ball is spun on the wheel to see in which slot it will land. The ball is equally likely to land in each slot. In the American Roulette there are 38 slots with the numbers 00, 0, and 1-36.

The board next to the wheel is used to place a bet for the next spin of the wheel and ball. If you place a bet on 17, then the ball would need to land on 17 to win, otherwise you lose your bet. If you place your bet on Even, then the ball would need to land on any even number for you to win, otherwise you lose your bet, and so forth. The three "2 to 1" spots allow you to bet the whole row of numbers, e.g. the bottom 2 to 1 would win if the ball lands on either 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, or 34. Betting on the spots "1 to 18" and "19 to 36" would win if the ball lands on any number 1 to 18 in the first case, and 19 to 36 in the second case.

There are many possible bets that can be made, each with its payout. For example, for a straight-up bet (meaning betting on a single number), the payout is 35 to 1, meaning if you bet $1 on a number and the ball lands on the number you selected, you will receive $35.

1. consider an inside bet (straight-up bet for example), and an outside bet (a 2 to 1 bet for example) and determine the probability for each (P(inside), P(outside));

2. determine the odds against each of your two bets and explain why these are not same as the payout for the same bets;

3. determine the probability of not winning for each of your two bets;

4. determine the expected value for each of your two bets separately if betting $1;