Consider a casino game?
a) Win-Win.......$8 gain......P = 0.0225 Win-Lose......$3 gain......P = 0.1275 Lose-Lose....$2 loss...... P = 0.7225 Lose-Win......$3 gain......P = 0.1275 b) Mean = $8(.0225) + $3(.1275) - 2(.7225) + 3(.1275) = -$0.50 (a loss of 50 cents) Variance = 0.0225(8 - (-0.50))^2 + 0.1275(3 - (-0.50))^2 + 0.7225(-2 - (-0.50))^2 + 0.1275(3 - (-0.50))^2 = 3.12375 Standard Deviation = Sqrt(3.12375) = 1.7674 c) If he plays 100 times, then his mean number of wins will be 100(0.15) = 15. In order to break even, Jack must win 20 games out of 100. We can use the normal distribution to approximate the probability of this with the statistic that np(1-p) is the variance of the binomial distribution. Variance in this case is 100(.15)(.85) = 12.75. Therefore standard deviation is 3.5707. Z = (20 - 15) / 3.5707 = 1.4 P(Z > 1.4) = 0.0808 <== 8.08 percent chance that he will have a positive profit. Win-Win: 0.15 x 0.15 = 0.0225 Win-Lose: 0.15 x 0.85 = 0.1275 Lose-Lose: 0.85 x 0.85 = 0.7225 Lose-Win: 0.85 x 0.15 = 0.1275 Total Probability is 1.
Many casinos have a game called the Big Six Money Wheel, in which a large wheel with various dollar amounts is?
Many casinos have a game called the Big Six Money Wheel, in which a large wheel with various dollar amounts is spun. Players may bet on one or more denominations; if the wheel stops on that denomination, the player wins that amount for each dollar bet. The wheel has 54 slots; the number of slots marked with each denomination is given in the following table. If a player bets $5 on the $20 denomination, find the player's expectation. (Round your answer to two decimal places.) $ ? Denomination Number of slots $40 2 $20 2 $10 4 $5 8 $2 15 $1 23
Does one poker chip equal one dollar?
White, $1 Yellow, $2 Red, $5 Blue, $10 Grey, $20 Green, $25 Orange, $50 Black, $100 Pink, $250 Purple, $500 Burgundy, $1000 Light Blue, $2000 Brown, $5000
Can a billionaire beat the house in any casino by constantly doubling down every 50% odds bet until he eventually wins?
Can a billionaire beat the house in any casino by constantly doubling down every 50% odds bet until he eventually wins?There needs to be a frequently asked questions list. This ought to be on it.In theory this should work, except … practicalities get in the way.It is true that in an indefinite sequence of games the probability of a win sometime is 1. When that occurs you are up by the amount of your original stake. Now start again. If you start with $1, your bets go $1, $2, $4, $8 and so on until you win. At that point you get back all your outlay plus $1. A person with a little over a billion dollars can afford 29 losses in a row but must win on the 30th play. However, 30 losses in a row sometime is also an event with probability 1. On average it will happen after about the same number of sequences that the billionaire will need to double his money. In other words, after winning one billion times there is a good chance that he will lose all that. There is a good chance that that will happen earlier (in which case the billionaire will be bankrupt), but it could also happen later.In practice the Casino might close for the night and the billionaire will lose his last accumulated bet. Also Casinos have betting limits to prevent their bankruptcy (although in this case they wouldn’t go bankrupt because they have already raked in the billionaire’s bet—they only stand to lose $1).Typically, if you bet on red at roulette, your probability of winning in Monte Carlo is 36/37 (in las Vegas it is 36/38—never play there). That’s almost 50:50 so on average it takes two spins of the wheel to make $1, so to make a meaningful amount of money the billionaire (say a 50% increase in his assets) will have to play about a billion games.
If a statistician had to bet $1,000,000 in a casino, how would they bet and why?
Assuming the statistician has no skill other than statistics (and therefore cannot play Poker or Blackjack to improve their return), and that their objective is to minimise losses and the variance associated with the loss, then the simplest strategy is to head to the Roulette Table.Divide the $1,000,000 by the number of squares on the table (usually 38 when there is a double zero, 37 in some European casinos with only a single zero). Place a $26,300 (or $27,000) bet simultaneously on each and every square (and "tip" the dealer the remaining few hundred dollars). You are guaranteed to win exactly 36 times your bet with which you leave the Casino.If the table has a bet limit, say $1,000, you may need to repeat the procedure of betting the limit on all squares a few times.Any other strategy at Roulette will increase the variance whilst leaving the expected return the same (as there is no bet that changes the expected return, whatever the strategy).Depending on the Casino and their particular rules you may be able to do better at, for example, the Craps table but the statistician would have to know more about the game. Roulette is by far the simplest and guaranteeing that the House is going to win [math]\tfrac2{38}[/math] or [math]\tfrac1{37}[/math] of a million in a single bet will probably smooth the statistician's passage.Note that the authorities will still be concerned that you are "washing" money obtained from some "dirty" source. Indeed the statistician should be worried about the source of the money themselves!