Class Notes for Probability & Statistics I (Math 3350) R. Sinn North Georgia College & State University August 25, 2009 Abstract A poker-based introduction to probability and statistics. Almost all the examples are real poker problems from No Limit Texas Holdem (NLHE) whose answers interest winning players. 1 Basic Probability 1.1 Organizing Principals Professional gamblers and poker players actually use mathematics to earn a living. Career paths for professional gamblers include running casinos and betting parlors in places where such things are legal. They o¤er odds for betting on athletic events and various races. They design and perfect games of chance for the casino ‡oor and verify that the games are fair. Well, some do. Professional poker players are currently the most high pro…le examples of professional gamblers. Ten years ago, few people outside Vegas knew of the legendary Doyle Brunson. Today, many of us can name poker celebrities like Phil Ivey, the "Poker Brat" Phil Hellmuth or Annie Duke having seen them playing high stakes poker on TV. What matematics do they use? They use probability theory to predict the future. They use statistics to analyzes the past. With an understanding of both, they manufacture hundreds of "small edges" over their competition to earn money. 1 A roulette ball falling into its slot and a card drawn from a well-shu• ed deck are two example of probability experiments. A probability experi- ment is any event whose outcomes are perfectly random and exactly pre- dicted by probabilities. The probability space (or sample space) is the col- lection of all possible outcomes of the experiment. A simple random sample (SRS) is a collection of outcomes drawn at random from the probability space S. An event is set of outcomes. We use set notation: an event A is a subset of S. Determining probabilities typically devolves into a counting excercise. We simply count the number of outcomes in event A and divide by the number of outcomes in S. To simplify our work, we will use the cardinality operator n(A) to indicate "the number of elements (outcomes) in set (event) A." This leads us to the only formula in discrete probability theory: n(A) P (A) = n(S) Example 1 Suppose we draw a single card at random from a well-shu• ed standard deck of 52 cards. What is the probability of drawing a heat? Let the event (set) H represent the 13 "hearts" outcomes. Then n(H) 13 1 P (A) = n(S) = 52 = 4 Generally, in probability caclulations, the compound event "A or B" means A B, while "A and B" indicates A B. Independence of the events causes[ a vital divergence. Sets A and B\ are independent provided A B = ?. \ 1. For indepedent sets: P (A B) = P (A) + P (B). [ 2. For dependent sets: P (A B) = P (A) + P (B) P (A B). [ \ 2 Note that when A B = ?, then by de…nition P (A B) = 0 since \ \ n(?) = 0. The cardinality of set with no elements is zero. We could about the probability of drawing one card and getting both an Ace and an 8. The events are certainly independent. But they can never occur simultaneously. For dependent sets, we typically have to …nd a creative way to count n(A B). Strategically, our counting job is much easier if we can partition the event\ into independent sets. When this is not possible, the formula is based on an idea called the "principal of inclusion-exclusion." The basic problem is that, when we sum, the outcomes in the "overlap" or intersection are counted twice. So we exclude them by subtracting. For three depedenet sets A, B and C, we have: P (A B C) = P (A) + P (B) + P (C) P (A B) [ P ([A C) P (B C) + P (A B C) \ \ \ \ \ We count all the two-way overlaps twice, and throw them out. In the process we "exclude" the three-way overlap three times - it was added three times - so now it has be included again. The same idea works regardless of how many sets are involved, but the results are tedious and in the long tradition of math texts are therefore left to the reader. It is often far easier to count the outcomes in the complement of event A, e.g. the probability that event A does not occur. We denote set com- plement as Ac. The formula is: c P (Ac) = n(A ) = 1 P (A) n(S) In probability calculations, we often …nd the notion of "throwing away" the overlap between two sets useful. This helps us identify independent sets. Given two events A and B such that A B = ?, the event "A but not B" is given by A Bc which we denote as A\ B6. \ n Example 2 Hero holds A J , and ‡op comes J 9 3 . Hero feels he will win with any club| (‡ush),| Jack (set), or Ace~ (two| pair),| but feels otherwise his hand is worse than his opponents. He believes he will lose unless his hand improves. What is his probability of making a winning hand on the next card? 3 Let A represent the event of a turning an Ace, and let C represent the event of turning a club. Since Hero holds the A , A C = ?, and thus P (A C) = P (A) + P (C). Hero can see four clubs,| the\ two in his hand and the two[ on the board. So n(C) = 9. There are three Aces left, so n(A) = 3. We know where …ve the 52 cards in the deck are, so: P (A C) = n(A)+n(C) = 3+9 = 12 [ n(S) 47 47 Example 3 Hero holds J T , and ‡op comes K Q 3 . Hero feels he will win with a straight, but~ worries} that Villain holds~ two| clubs| (‡ushes beat straights). What is his probability of making the winning hand on the next card? Let T be the event Hero turns a straight card (any A or 9), and let F be the event a ‡ush card hits (any club). Here, the winning event for Hero is T F , and n(T F ) = 8 2, that is the 8 straight cards from which we "thrown away" the An and the 9 . The solution is 6 . | | 47 Example 4 Hero faces one villain heading to the ‡op where Hero bets last. It is known that the probability of "hitting the ‡op" with any two unpaird cards is approximately 1/3. Hero decides to blu¤ his opponent. What is the probability his opponent "missed" the ‡op and will therefore be likely to fold? Let H be the event of "hitting the ‡op," e.g. pairing one (or more) of the board cards. The important probability is knowing that P (not H) = 1 2 1 3 = 3 . Aggressive poker players use this likelihood of "unimproved" cards in villain’shand to bet (blu¤) on the ‡op. This is so common it has a name: a continuation bet, or cbet. 4 1.1.1 Odds One of the more confusing aspects of Vegas-style betting is that gamblers tend to state probabilities in terms of odds rather than percentages. A weatherman would say "there’s a 70% chance of rain tomorrow." A gam- ber would lay 7 to 3 odds for rain tomorrow. A percentage probability is 7 based on the fraction, here, 10 . In fractional probabilities, we give the "num- ber of successes" divided by the "total outcomes." For odds, we give a ratio of "number of successes" to "number of failures." The order is important: success to failure. Poker players know a host of odds. For example, Hero faces odds of roughly 7.5 to 1 against ‡opping set when starting with a poket pair (pp). The "against" turns the odds around, meaning there is likely to be "1" success for each "7.5" failures. To convert to a probability, we …nd the denominator by …rst adding successes and failures. Let FS be the event of ‡opping a set given that Hero starts with a pp. Then: 1 1 2 P (FS) = 7:5+1 = 8:5 = 17 = 0:117 65 Suppose Hero holds TT while his opponent holds 55. Hero is about a 4 to 1 favorite. This means that a hand like TT will beat a hand like 55 about 4 times in 5, or 80% of the time. Poker players tend to use both percentage chance of winning (called eq- uity) and odds, depending upon the situation. When betting, knowing one’s odds is more useful since it allows us to match our bet-to-pot ratio and com- pare it to our successes-to-failure ratio. For example, Hero estimates his chances of winning the pot are 1 in 3, or 2 to 1 against. Villain bets $50 into a $100 pot. Hero must call $50 for a chance to win $150. Hero has 2 to 1 odds (against) making his hand, but the pot is "laying him" 3 to 1 odds ($150 to $50). In this case, it is best for Hero to call, since the odds against him making his hand are lower than the pot odds. 1.1.2 Equity Once you understand the ratio vs. fraction concept, converting odds to probabilities is pretty simple and can usually be at least estimated in your 5 head. Going from percentages to odds can be a bit more tricky. Poker players tend to use various odds calculators like PokerStove (highly recommended, free – google "Poker Stove").