Yahtzee Rules and Scoring, Study notes of Probability and Statistics

Download Yahtzee Rules and Scoring and more Study notes Probability and Statistics in PDF only on Docsity! Yahtzee Yahtzee is a dice game based on Poker. The object of the game is to roll certain combinations of numbers with five dice. At each turn you throw dice trying to get a good combination of numbers; different combi- nations give different scores. While luck plays a big role in Yahtzee, strategy makes a significant difference. The reason for this is that you score each combination just once, and the number of different combinations in which you can score is equal to the number of turns in the game. This means that you have to make wise choices about when to score in each combination and you have to be careful about what combinations you seek at each turn. Rules and Scoring The game takes thirteen turns. Each turn consists of up to three separate rolls of the dice. On the first roll you roll all five of the dice. After the first and second roll, you can hold any subset of the five dice you want (including none of the dice or all of the dice) and roll the rest trying to get a good combination. After the three rolls (or after the first or second roll if you choose to stop) you must find a place among the thirteen boxes in the scorecard to put your score. The score you get depends on the box that you choose and the combination that you have rolled. After a box is used, you can’t use it again, so you have to choose wisely. This means that, in general, you don’t have to choose the box that gives you the highest score for the combination you have rolled, since it may be advantageous to save that box for an even better roll later in the game. In fact, there are many situations it which it makes sense to put a 0 in a “bad” box instead of a low score in another “good” box because doing so would block the good box for future turns. The Yahtzee scorecard contains two sections: the upper section and the lower section. The Upper Section In the upper section, there are six boxes, each corresponding to one of the six face values of the dice. For these boxes, you enter the sum of the dice with the corresponding face value (and ignore all other dice). For example, if you roll 5 3 3 5 3 you may enter a score of 9 in the third spot or a score of 10 in the fifth spot. Bonus: At the end of the game, you get a bonus of 35 points if the total number of points you scored in the upper section is 63 or higher. The seemingly random number 63 corresponds to having scored a combination with three dice of the corresponding face value in each of the upper section boxes (though any 1 upper section total of 63 or greater is rewarded with the bonus). The Lower Section Here you can score for various combinations. • 3 of a kind: Score in this box if the dice include 3 or more of the same number. The score is the total of all five dice (unlike the upper section, where only the dice of one number are summed). • 4 of a kind: Score in this box if the dice include 4 or more of the same number. The score is the total of all five dice. • Full House: Score in this box if the dice show three of one number and two of another (for example, 2 6 6 2 2). The score is 25 points. • Small Straight: Score in this box if the dice show any sequence of 4 numbers (for example 3 2 4 4 5). The score is 30 points. • Large Straight: Score in this box if the dice show any sequence of 5 numbers (for example 5 2 4 6 3). The score is 40 points. • Chance: Score in this box with any combination. The score is the total of the five dice. • Yahtzee: Score in this box if the dice show five of the same number. The score is 50 points. • Yahtzee Bonus: If you roll a second Yahtzee and the Yahtzee box is already filled with 50 points, you get a 100 points bonus. If you roll a Yahtzee and your Yahtzee box is filled with a zero, you do not receive this bonus. In either case, you must fill one of the other boxes on your scorecard using the “joker rules” as follows: if possible, you must fill the upper section box corresponding to the number you rolled. If that box is already filled, you must fill in any open lower section box. For the 3 of a kind, 4 of a kind, and chance boxes, this means scoring the total of all 5 dice. For the full house, small straight, and large straight boxes, this means scoring 25, 30, or 40 points respectively (even though you did not actually roll the corresponding combination). If the entire lower section is filled you must enter a zero in any open box in the upper section. Basic Tips The following are generally regarded as the basic things to keep in mind while playing Yahtzee. • You should generally aim for the 35 points bonus by filling the upper section with high scores (so as to get at least 63 points) near the beginning of the game. • If you roll a combination for which your available boxes give a low score, fill in a zero in the 1’s box (or even the 2’s or 3’s), since high scores in the other boxes may compensate for this. 2 6. Suppose you are playing a game where you can roll a single die one or two times. Your objective is to get a high number. After the first roll you can choose whether to keep the number you got or re-roll. Then the best strategy is to keep a 4, 5, or 6, while re-rolling a 1, 2, or 3 (try to convince yourself of this). Now suppose we play the same game but with the option of rolling the single die one, two, or three times. As before, after the second roll you should keep a 4, 5, or 6, and re-roll a 1, 2, or 3. Can you come up with a strategy on what to do after the first roll such that the expected (or average) score you get in the three rolls game is better than the expected score in the two rolls game with the strategy we proposed? What is the expected score when playing with your proposed strategy? These types of calculations are useful in the actual Yahtzee game, for example when you are trying to score well in the chance box. 7. It is the last roll of the game and you need a Yahtzee to win. After the first roll you keep the dice of the most common number you rolled and then re-roll twice to see if you can end up with all of that number. (We ignore the possibility that you change the number for which you are looking after the first roll, e.g., you only had one 5 after the first roll and then rolled two 4’s.) In the case of a tie on the first roll, choose one of the numbers that appears the maximal number of times and keep only the dice with that number. What is the probability you end up with a Yahtzee? This problem is rather difficult, so don’t expect to get a fast answer. Here is a hint: first compute the probability of getting 2, 3, 4, or 5 of a kind, or all different numbers in the first roll; then try to compute the probability of passing from having i of a kind in a roll to having j on the next roll and try to use these “transition probabilities”. 8. In reality, if you are going for a Yahtzee and you keep two 4’s after you first roll but roll three 5’s on your second roll, you will try for a Yahtzee in 5’s on your last roll. Repeat the previous problem, but now incorporating the fact that after your second roll you may change the number that you are seeking. Once again, if there is a tie, choose one number to keep. If you get all different numbers on your first roll, re-roll all of the dice. 5 Solutions 1. Here it is easier to find the probability of not getting a six. Each time you roll, the probability of not getting a six is 5 6 , thus the probability of not getting a six at all is ( 5 6 )3 = 125 216 . So the probability of getting at least one six is 1 − 125 216 = 91 216 , or about 0.4213. 2. On the first roll you will fail to get a 1 or 6 with probability 4 6 = 2 3 . The probability you fail twice is then ( 2 3 )2 = 4 9 , so you succeed with probability 1 − 4 9 = 5 9 = 0.5555. 3. On the first roll you will fail to get a 4 with probability 5 6 . The probability you fail twice is ( 5 6 )2 = 25 36 so you succeed with probability 1 − 25 36 = 11 36 or about 0.3056. So the probability of succeeding in this case is lower than in the previous case. This is because here you are looking for an “inside” straight, so only one number works, while in the previous problem you were looking for an “outside” straight and had two numbers that would work. 4. The probability we do not roll a six in two tries with one die is (5/6)2, so the probability that we do not roll a 6 when we roll three dice twice each is ( 5 6 )6 ≈ 0.3349. Therefore the probability of rolling at least three 6’s is 1 − 0.3349 = 0.6651. To finish with exactly four sixes we need to be successful (i.e. roll a 6 in two tries) with two dice and fail with the one other die. By the binomial distribution, this occurs with probability 3 · ( 11 36 )2 · 25 36 ≈ 0.1945. For a Yahtzee we must be successful with all 3 dice which occurs with probability ( 11 36 )3 ≈ 0.0285. 5. There are 62 = 36 possible outcomes when we re-roll the 1 and the 4. We first calculate the number of different outcomes that result in a particular hand and use this to determine the probability of each hand. For example, the probability of rolling a 2 and a 6 is 2/36 since we could roll the 2 and then the 6, or the 6 and then the 2. The probabilities of each of the hands are: Hand Number of Outcomes Probability Straight 6 - 2 2 2/36 = 1/18 Large 1 - 2 2 2/36 = 1/18 Large 6 - (not 2) 4 · 2 + 1 = 9 9/36 = 1/4 Small 2 - (not 1 or 6) 3 · 2 + 1 = 7 7/36 Small no 2 or 6 42 = 16 16/36 = 4/9 None Note that a six paired with any other number can occur two ways, but a pair of sixes occurs in only one way. This is why the third and fourth rows have a plus one in the number of outcomes column. Next, for each of the possible first roll outcomes we calculate the probability of getting a large straight or a small straight on the third roll. For the 6-2 and 1-2 hands we already have our large straight so we are done. For the 6 - (not 2) hand we re-roll the non-2 die. We have a 1/6 chance of getting a two and a large straight, and a 5/6 chance of not getting a 2 and having only a small straight. For the 2 - (not 1 or 6 hand) we will keep the 2,3,4,5 and re-roll one die, giving us a 1/3 chance at rolling a large straight and a 2/3 chance of having only a small straight. In the case where we rolled no 2 or 6 on 6 the second roll, the situation for our third roll is the same as the situation for our second roll, so the above table gives the probabilities of each outcome. Combining these results we see that the probability of a large straight is 1 18 + 1 18 + 1 4 · 1 6 + 7 36 · 1 3 + 4 9 · ( 1 18 + 1 18 ) ≈ 0.267. The probability of a small straight (but not a large straight) is 1 4 · 5 6 + 7 36 · 2 3 + 4 9 · ( 1 4 + 7 36 ) ≈ 0.535. The probability of getting no straight at all is 4 9 2 ≈ 0.198. If we re-roll only one die, then on our second roll we have a 1/6 chance of rolling a 2 and getting a large straight. We also have a 1/6 chance of getting a 6 and a small straight. In this case, on our third roll we have a 1/6 chance of rolling a 2 and getting a large straight. However, 2/3 of the time we get neither a 2 or a 6 on our first roll in which case our situation does not change. Therefore our chance of getting a large straight is 1 6 + 1 6 · 1 6 + 2 3 · 1 6 ≈ 0.306. The probability of getting only a small straight is 1 6 · 5 6 + 2 3 · 1 6 = 0.25. The chance of getting no straight is 2 3 2 = 4 9 ≈ 0.444. So re-rolling only the 4 gives you a slightly higher chance of getting the large straight, but a much lower probability of at least ending up with a small straight. If you only need a large straight then re-rolling just the four is the better strategy, but if a small straight would also be valuable then re-rolling the 1 and the 4 might be a better move. 6. The strategy we proposed for the two rolls game has the following expected score: 4 · 1 6 + 5 · 1 6 + 6 · 1 6 + 3.5 · 1 2 = 4.25 Now for the three rolls game, using the same idea as in the two rolls game, we could use the following strategy: after the first roll: keep a 5 or a 6 and re-roll a 1, 2, 3, or 4. The expected score using this strategy is: 5 · 1 6 + 6 · 1 6 + 4.25 · 2 3 = 14 3 ≈ 4.67 (the last term in the sum comes from the fact that, if you re-roll after the first roll, then you are back in the situation of the two rolls game, so your expected score is 4.25, as before). So this strategy for the three rolls game has a better expected score than the strategy for the two rolls game. 7. Recall that the number of ways of picking m things out of n is Cn,m = n! m!(n − m)! = n(n − 1) · · · (n − m + 1) 1 · 2 · · ·m 7