A major mathematical technique used throughout baseball these days is probability. Probability influences baseball more than any other sport since baseball can be broken down into a sequence of one on one battles – the pitcher vs. the batter. Baseball analysts use probability to forecast which teams will make the postseason, to determine the likelihood that a player will hit a home run or bat around a runner, or even what the next pitch could be. To understand these complexities, we must decipher the differences between independent and dependent events. Without further ado, let us dive into a realm where probabilities have been utilized for decades if not centuries, the casino. One of our favorite casino games to analyze under a probability lens is roulette. Not only is it a popular casino staple, but is also often declared as the oldest casino game still in play today. The first accepted reference is as a byproduct of an attempted perpetual motion machine invented by French mathematician Blaise Pascal in the late 1640’s, but roulette in its current form did not appear until 1796 in Paris casinos. Although there are two major versions of the game, for this analysis I am going to use the American version of the wheel. For those of you unfamiliar with its layout, the American roulette wheel has 38 slots numbered 00, 0, and 1 through 36. Here are the color distributions across the numerals:
The numbers on the wheel are ordered in such a way that high and low numbers, as well as odd and even numbers, alternate in addition to red and black. The game then proceeds in the following way. First, the wheel is spun and a small ball is subsequently rolled into the groove, in the opposite direction as the motion of the wheel. Eventually the ball falls into one of the slots. If the slot matches a bet, then the player wins. Although there are many types of bets that can be placed, let me reiterate the classic ones:
Caution: Math Ahead! Since we can assume that the wheel is fair, a random variable X of the slot numbers can be constructed using the discrete uniform distribution. The sample space simply becomes every possible slot number, S = {00, 0, 1, 2, 3, 4… , 36}. The discrete uniform distribution has a probability mass function of 1/N. Using N = 38 = number of slots, we can confidently say that the probability for an element x in S is 1/38, or, P(X =x) = 1/38 for each x in S. Now that we know the probability mass function for a roulette wheel, we can prove the independence of two separate rolls of the ball, or two succeeding events for any combination of bets. The equation for independence of two events that I am going to use stems from conditional probability. Two events are independent if and only if P(A & B) = P(A) * P(B). Here is our sample space, S2, for the possible amount of slot numbers to be selected per bet: S2 = {1, 2, 3, 4, 6, 12, 18}. So for any two bets y1, y2 in S2, let P(A) be the event of winning y1 on the first roll and P(B) be the event of winning y2 on the second roll. P(A & B) then equals (y1 * y2)/1444 = y1/38 * y2/38 = P(A) * P(B). Thus P(A) and P(B) are independent. We can extend this to any number of rolls, with one bet per roll. Here, let n be the total number of rolls and yi be the number of slots selected per bet, as defined in S2. Thus let the probability of winning every bet be defined as: which is equivalent to the product of the probabilities of each individual bet success.
It is interesting to note that successes of different bets in the same roll are NOT always independent events. This can be obviously seen in the case where Player 1 selects slot 21 and Player 2 selects slot 33. The probability of individual success is 1/38 in each case, however the joint probability of them both winning is zero. Since the ball can only land in one slot, there is no scenario where they both can win, so P(A & B) != P(A) * P(B). A more indepth example regards the two simultaneous bets of the ball landing in a Low slot, and also a Red slot. Both have individual probabilities of 18/38. Thus P(A) * P(B) is approximately 324/1444 = .224. However, P(A & B), as we can discern from above, is actually 9/38 which is approximately .237. In conclusion, while separate successful bets within one roll of the ball are not always independent events, each individual roll of the ball is independent of any roll that comes before or after it. This allows roulette to be analyzed as a random walk, resulting in a popular gambling strategy called The Martingale. we won’t go in depth here, but it does not work as a player would need unlimited wealth and casinos have maximum bets for a reason (to guarantee profit!). The independence of rolls also disproves the Gambler’s fallacy, where the fact that the ball has just hit black 26 times in a row, does not prove that the ball will hit red on its next go around the wheel. Finally, here is a great story to end with on how someone bet all of his life savings on red and walked away with double his money. The SaberSmart Team
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