just think about mathematics language
Problem 2 In Yahtwentee, players roll five 20-sided dice at a time. (The values on a twenty-sided die are 1 through 20, inclusive.)
a) A “full house” is rolled when three of the dice have the same value (these three form a “triplet”) and the remaining two dice have the same value as each other (these two form a “pair”) but a different value than the dice in the triplet. Calculate the probability that a player rolls a full house in their first roll in Yahtwentee.
b) In Yahtzee, players roll five 6-sided dice in each turn. Calculate the probability that a player rolls a full house in their first roll in Yahtzee.
c) Polyhedral dice come in 4-sided, 6-sided, 8-sided, 12-sided, and 20-sided variants. Suppose you get to pick how many sides the five dice you roll will have. Which kind of dice should you use to maximize the probability of rolling a full house?
Problem 3 The numbers of items with various traits in a second-hand shop were tallied: 35 were valuable; 40 were antiques; 22 were rare antiques; 16 were valuable antiques; 20 were rare and valuable; 11 were rare, valuable antiques; 30 were neither rare, nor valuable, nor antiques; There were 100 items total.
a) How many rare items are in the shop? Draw a Venn diagram to justify your answer.
b) I purchase two items at random from the shop. What is the probability that both of them are valuable antiques?
Problem 4 A group of researchers wants to understand coffee consumption.
a) The researchers began by studying the relationship between age and coffee consumption. They surveyed 20 young adults (average age 24 years old), 60 middle-aged adults (average age 45 years old), and 20 elderly adults (average age 70 years old). Of those surveyed, 10 young adults said they drink coffee, 40 middle-aged adults said they drink coffee, and 5 elderly adults said they drink coffee. Determine the mean age of a coffee drinker in the study.
b) Assuming that coffee drinkers each drink precisely one cup of coffee each day, find the probability that a cup of coffee consumed by a participant in the study was consumed by a middle-aged adult.
c) The researchers were also interested in how people prepared coffee. There were three techniques used by participants: drip, pour over, and instant. They found that people only used one technique each day, and that each person’s choice of technique each day depended only on the technique they used the day before. If someone used drip one day, they would use drip 80% of the time and pour over 20% of the time the following day. If they used pour over one day, then the following day they used drip 30% of the time, pour over 50% of the time, and instant 20% of the time. If they used instant, they used drip 10% of the time and instant 90% of the time the following day. Draw a transition diagram depicting the researchers’ findings.
d) Explain why this is a regular Markov chain.
e) On the first day the researchers surveyed the participants, they found that 75% had used drip, 10% had used pour over, and 15% had used instant. Determine the distribution of coffee preparation techniques used amongst the participants after an arbitrarily long time has passed.
