![]() (For reasons why it is difficult to bias a coin-flip in practice, see e.g., Gelman and Nolan 2002 and Diaconis, Holmes and Montgomery 2007.) Now, suppose you were to execute a sequence of coin tosses in such a way that you start off biasing your tosses towards heads, but each time the sample proportion of heads exceeds 60% you change to bias towards tails, and each time the sample proportion of tails exceeds 60% you change to bias towards heads. Suppose that this method is sufficiently effective that you can bias the coin 70-30 in favour of one side. What if exchangability doesn't hold? Can there be a lack of convergence? Although there are also weaker assumptions that can allow similar convergence results, if the underlying assumption of exchangeability does not hold -i.e., if the probability of a sequence of coin-toss outcomes depends on their order- then it is possible to get a situation where there is no convergence.Īs an example of the latter, suppose that you have a way of tossing a coin that can bias it to one side or the other -e.g., you start with a certain face of the coin upwards and you flip it in a way that gives a small and consistent number of rotations before landing on a soft surface (where it doesn't bounce). (This is due to a famous mathematical result called de Finetti's representation theorem see related questions here and here.) The strong law of large numbers then kicks in to give you the convergence result -i.e., the sample proportion of heads/tails converges to the (fixed) probability of heads/tails with probability one. Now, if this assumption holds then the sequence of outcomes will be IID (conditional on the underlying distribution) with fixed probability for heads/tails which applies to all the flips. The assumption of exchangeability is the operational assumption that reflects the idea of "repeated trials" of an experiment - it captures the idea that nothing is changing from trial-to-trial, such that sets of outcomes which are permutations of one another should have the same probability. For example, the condition of exhangeability would say that the outcome $H \cdot H \cdot T \cdot T \cdot H \cdot T$ has the same probability as the outcome $H \cdot H \cdot H \cdot T \cdot T \cdot T$, and exchangeability of the sequence would mean that this is true for strings of any length which are permutations of each other. If the coin is tossed in a manner that is consistent from flip-to-flip, then one might reasonably assume that the resulting sequence of coin tosses is exchangeable, meaning that the probability of any particular sequence of outcomes does not depend on the order those outcomes occur in. The convergence outcome follows from the condition of exchangeability. Your bet is subtracted from your score.This is an excellent question, and it shows that you are thinking about important foundational matters in simple probability problems. Two dice – You win! Bet points are added to your score. Because you only have two dice and the Devilkin have three, ties are in your favor. Whoever has the highest die wins each comparison. In this game of chance, you roll dice with the Devilkin.Īfter your roll, the highest numbered dice from each set are compared. This mini-game is currently only available on the Facebook version of Slingo® Adventure. You have a 50/50 chance of winning the bet. The Devil will choose one of the remaining options. In this minigame, select your bet and then choose rock, paper, or scissors. Follow the shell that hides the ball and win! The Joker's Shell Game is more of a skill game. ![]() The Devil's Coin Toss is a simple win or lose game. You can choose to not play a minigame by spending coins or by using the Minigame Shield Power Up before having the minigame appear. In these cases, no points are awarded or taken away. It's possible to draw (neither win or lose) on some minigames. Winning will award points, losing will remove them. To participate, just choose the percentage of points you’d like to bet: 10%, 25%, or 50%. When a die comes up in the reels, it’s time for a minigame. Minigames are quick, high-stakes games that occur at random during play.
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