Roulette is a game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even. The 18 red, 18 black, and 2 green pockets of the roulette wheel are numbered from 1 to 36, 0, and 00. The payout is 1 to 1 if you bet on red or black, 1 to 1 if you bet on even or odd numbers, 35 to 1 if you be on one number among 38 available numbers.
1) Daniel decides to bet on the number "00" and will stop when he wins twice:
a) What is the average of the games that Daniel expects to play?
b) Determine the probability that Daniel will stop before 8th game.
c) Determine the probability that Daniel will stop before losing 8 games.
2) What is the probability that there will be no more than 8 games that the ball falls into a red pocket in the following 10 games.
3) Determine the mean and variance of the number of games that the ball falls into a odd-numbered pocket in the next 30 games.
4) Suppose there are records of 120 games and the ball fell into a black-coloured pocket 50 times. In a random sample of 10 games (sampling without replacement) from these 120 records, what is the probability that we sample at most two games with black winning colour.
5) Records show that 87,600 games have played last year and the ball fell into a black- coloured pocket 41,855 times. In a random sample of 10 games from these records, what is the probability that we sample at most two games with black winning colour.