## Question 4: notorious discrete probability distributions in R a. To maximize their time sleeping, many commuter students leave at the last possible moment for early-morning classes. Most of the time, traffic and light timings work out so that they arrive on time. Occasionally, they are late. Imagine that the probability of making it on time is 78%, and that the outcomes are independent from day to day. The number of days someone is on time, out of the 36 days they travel to their early-morning class, can be modeled with a binomial distribution. Find the probability that: * they are on time 32 or more days; about 7.8% * they are on time every day; about 0.013% * they are on time between 25 and 30 days (inclusive); about 75.5% {r Q4a} # 32 or more days #Every day #Between 25-30 inclusive b. Let's model the number of times that a student purchases food in the student union (over the course of a semester) with a Poisson distribution with an average of 3.2. Determine the value of `lambda` for this Poisson (hint: this is an easy one), then find: * the probability someone makes 5 or more purchases (about 21.9%) * the probability someone makes exactly 8 purchases (about 1.1%) * the probability someone makes the average number of purchases {r Q4b} #5 or more purchases #8 purchases #Average number of purchases C. You are asked to give out flyers about the Master's in Business Analytics program to people around Haslam. Imagine that each person, independently, has a 4.2% chance of accepting the flyer. After a while, you have only a single flyer to give away. The geometric distribution provides a model for the number of people you must ask *before* someone takes the last flyer. For the following questions, figure out what it's asking about the number of "failures" (refusals of the flyer) before the first "success" (accepting the flyer), then find the requested probabilities. * the probability that you will give away the last flyer to the 10th person you ask (about 2.9%). * the probability that you'll give away your last flier to the 11th, 12th, 13th, 14th, or 15th person you ask (about 12.6%) * the probability that you will have to ask 25 or more people to give away your last flyer (about 35.7%) ``{r Q4c} # Last flyer to the 10th person you ask #Last flyer to person 11-15 inclusive #At least 25 people d. Right after you give away your last flyer, you're handed 6 more. Find the probability that: * you have to ask 120 or more people to give them all away (about 61.6%) * you have to ask at least 130 but no more than 160 people to give them all away (about 20.9%) Remember that the negative binomial counts up the number of *failures* before a certain number of successes. A failure is someone refusing a flyer. A success is someone accepting a flyer. Translate these questions into how many failures are required to describe the event, then use the `nbinom suite of functions. `{r Q4d} #Ask 120 or more people #At least 130 but no more than 160