A Collection of Dice Problems - madandmoonly.com

1 A Collection of dice Problemswith solutions and useful appendices(a work continually in progress)version December 2, 2020 Matthew M. Conroydoctormatt at madandmoonly dot Collection of dice ProblemsMatthew M. ConroyThanksA number of people have sent corrections, comments and suggestions, including Ryan Allen, Julien Beasley,Rasher Bilbo, Michael Buse, Stephen B, Paul Elvidge, Amit Kumar Goel, Steven Hanes, Nick Hobson,Marc Holtz, Manuel Klein, David Korsnack, Peter Landweber, Jason Cheuk-Man Leung, Paul Micelli,Albert Natian, Jo ao Neto, Khizar Qureshi, Michal Stajszczak, Dave TeBokkel, Yichuan Xu and Elie , 1 Introduction and NotesThis is a (slowly) growing Collection of dice -related mathematical Problems , with accompanying solu-tions. Some are simple exercises suitable for beginners, while others require more sophisticated dice Problems have an advantage over some other Problems of probability in that they can beinvestigated experimentally.

2 This gives these types of Problems a certain helpful down-to-earth feel free to comment, criticize, or contribute additional What are dice ?In the real world, dice (the plural ofdie) are polyhedra made of plastic, wood, ivory, or other hardmaterial. Each face of the die is numbered, or marked in some way, so that when the die is cast onto asmooth, flat surface and allowed to come to rest, a particular number is , we can consider a die to be a random variable that takes on only finitely many distinctvalues. Usually, these values will constitute a set of positive integers1,2,..,n; in such cases, we will referto the die TerminologyAfairdie is one for which each face appears with equal likelihood. A non-fair die is calledfixed. Thephrasestandard diewill refer to a fair, six-sided die, whose faces are numbered one through six. If nototherwise specified, the termdiewill refer to a standard Standard Dice1. On average, how many times must a6-sided die be rolled until a 6 turns up?

3 2. On average, how many times must a6-sided die be rolled until a 6 turns up twice in a row?3. On average, how many times must a6-sided die be rolled until the sequence 65 appears ( , a 6followed by a 5)?4. On average, how many times must a6-sided die be rolled until there are two rolls in a row that differby1(such as a2followed by a1or3, or a6followed by a5)? What if we roll until there are tworolls in a row that differ by no more than1(so we stop at a repeated roll, too)?5. We roll a6-sided dientimes. What is the probability that all faces have appeared?6. We roll a6-sided dientimes. What is the probability that all faces have appeared in order, in somesix consecutive rolls ( , what is the probability that the subsequence123456appears among thenrolls)?7. Person A rollsndice and person B rollsmdice. What is the probability that they have a commonface showing ( , person A rolled a 2 and person B also rolled a 2, among all their dice )?

4 8. On average, how many times must a6-sided die be rolled until all sides appear at least once? Whatabout for ann-sided die?9. On average, how many times must a6-sided die be rolled until all sides appear at leasttwice?10. On average, how many times must a pair of6-sided dice be rolled until all sides appear at least once?11. Suppose we rollndice. What is the expected number of distinct faces that appear?12. Suppose we rollndice and keep the highest one. What is the distribution of values?13. Suppose we can roll a6-sided die up tontimes. At any point we can stop, and that roll becomes our score . Our goal is to get the highest possible score, on average. How should we decide when tostop?14. How many dice must be rolled to have at least a 95% chance of rolling a six?4A Collection of dice ProblemsMatthew M. Conroy15. How many dice must be rolled to have at least a 95% chance of rolling a one and a two? What abouta one, a two, and a three?

5 What about a one, a two, a three, a four, a five and a six?16. How many dice should be rolled to maximize the probability of rolling exactly one six? two sixes?nsixes?17. Suppose we roll a fair die 100 times. What is the probability of a run of at least 10 sixes?18. Suppose we roll a fair die until some face has appeared twice. For instance, we might have a run ofrolls 12545 or 636. How many rolls on average would we make? What if we roll until a face hasappearedthreetimes?19. Suppose we roll a fair die 10 times. What is the probability that the sequence of rolls is non-decreasing( , the next roll is never less than the current roll)?20. Suppose a pair of dice are thrown, and then thrown again. What is the probability that the facesappearing on the second throw are the same as the first?What if three dice are used? Or six?21. A single die is rolled until a run of six different faces appears. For example, one might roll the se-quence 535463261536435344151612534 with only the last six rolls all distinct.

6 What is the expectednumber of rolls?22. What is the most probable: rolling at least one six with six dice , at least two sixes with twelve dice ,or at least three sixes with eighteen dice ? (This is an old problem, frequently connected with IsaacNewton.)23. Suppose we rollndice, remove all the dice that come up 1, and roll the rest again. If we repeat thisprocess, eventually all the dice will be eliminated. How many rolls, on average, will we make? Show,for instance, that on average fewer thanO(logn)throws Suppose we roll a die6ktimes. What is the probability that each possible face comes up an equalnumber of times ( ,ktimes)? Find an asymptotic expression for this probability in terms Call a consecutive difference the absolute value of the difference between two consecutive rollsof a die. For example, the sequence of rolls14351has the corresponding sequence of consecutivedifferences3,1,2,4. What is the expected number of times we need to roll a die until all6consecutivedifferences have appeared?

7 dice Sums26. Show that the probability of rolling 14 is the same whether we throw 3 dice or 5 dice . Are there otherexamples of this phenomenon?27. Show that the probability of rolling a sum of9with a pair of5-sided dice is the same as rolling a sumof9with a pair of10-sided dice . Are there other examples of this phenomenon? Can we prove thereare infinitely many such?28. Suppose we rollndice and sum the highest3. What is the probability that the sum is 18?5A Collection of dice ProblemsMatthew M. Conroy29. Four fair, 6-sided dice are rolled. The highest three are summed. What is the distribution of the sum?30. Three fair,n-sided dice are rolled. What is the probability that the sum of two of the faces rolledequals the value of the other rolled face?31. A fair,n-sided die is rolled until a roll ofkor greater appears. All rolls are summed. What is theexpected value of the sum?32. A pair of dice is rolled repeatedly. What is the expected number of rolls until all eleven possible sumshave appeared?

8 What if three dice are rolled until all sixteen possible sums have appeared?33. A die is rolled repeatedly and summed. What can you say about the expected number of rolls untilthe sum is greater than or equal ton?34. A die is rolled repeatedly and summed. Show that the expected number of rolls until the sum is amultiple A fair,n-sided die is rolled and summed until the sum is at leastn. What is the expected number ofrolls?36. A die is rolled and summed repeatedly. What is the probability that the sum will ever be a given valuex? What is the limit of this probability asx ?37. A die is rolled and summed repeatedly. Letxbe a positive integer. What is the probability that thesum will ever bexorx+ 1? What is the probability that the sum will ever bex,x+ 1, orx+ 2? A die is rolled once; call the resultN. ThenNdice are rolled once and summed. What is thedistribution of the sum? What is the expected value of the sum?

9 What is the most likely value?What the heck, take it one more step: roll a die; call the resultN. RollNdice once and sum them;call the resultM. RollMdice once and sum. What s the distribution of the sum, expected value,most likely value?39. A die is rolled once. Call the resultN. Then, the die is rolledNtimes, and those rolls which areequal to or greater thanNare summed (other rolls are not summed). What is the distribution of theresulting sum? What is the expected value of the sum?40. Supposensix-sided dice are rolled and summed. For each six that appears, we sum the six, and rerollthat die and sum, and continue to reroll and sum until we roll something other than a six with that is the expected value of the sum? What is the distribution of the sum?41. A die is rolled until all sums from1toxare attainable from some subset of rolled faces. For example,ifx= 3, then we might roll until a1and2are rolled, or until three1s appear, or until two1s and is the expected number of rolls?

10 42. How long, on average, do we need to roll a die and sum the rolls until the sum is a perfect square(1,4,9,16,..)?43. How long, on average, do we need to roll a die and sum the rolls until the sum is prime? What if weroll until the sum is composite?6A Collection of dice ProblemsMatthew M. Non-Standard Dice44. Show that the probability of rolling doubles with a non-fair ( fixed ) die is greater than with a fair Is it possible to have a non-fair six-sided die such that the probability of rolling2,3,4,5,and6is thesame whether we roll it once or twice (and sum)? What about for other numbers of sides?46. Find a pair of 6-sided dice , labelled with positive integers differently from the standard dice , so thatthe sum probabilities are the same as for a pair of standard Is it possible to have two non-fairn-sided dice , with sides numbered1throughn, with the propertythat their sum probabilities are the same as for two fairn-sided dice ?48. Is it possible to have two non-fair 6-sided dice , with sides numbered1through6, with a uniform sumprobability?