Nested Problems Chris Morgan, MATH G160 csmorgan@purdue.edu February 15, 2011 Lecture 16

Nested Problems In each of the following cases, name the distribution that can best be used to describe the random variable X and give the value of all necessary parameters for that distribution.

Nested Example #1a You have a bag of 100 M&M’s. 30 are green. If you grab a handful of 10 M&M’s, let X be the number of green M&M’s you grab. X ~ Hyp(N=100, n=10, p=0.3)

Nested Example #1b Next you ask your neighbor if they like M&M’s. Assume that 70% of all people like M&M’s. Let X=1 if they say yes and X=0 if they say no. X ~ Ber(p=0.7)

Nested Example #1c Again, assume that 70% of all people like M&M’s. In a class of 40 students, let X be the number of students who like M&M’s X ~ Bin(n=40, p=0.7)

Nested Example #1d Suppose you have a jumbo bag of 10,000 M&M’s and 500 of them are yellow. Let X be the number of yellow ones you get in a handful of 10 M&M’s. Exact: X ~ Hyp(N=10000, n=10, p=0.05) Approximate: X ~ Bin(n=10, p=0.05)

Nested Example #1e On average, 3 people in the U.S. are diagnosed with toxic M&M’s overdose syndrome per month. Let X be the number of people diagnosed with this symptom next year. X ~ Poi(λ=36)

Nested Example #2 The confectionery company Chocoholly makes chocolate chip cookies as part of their production line. Chocolate chips in the cookies are randomly and independently distributed according to a Poisson distribution with an average of 12 chocolate chips per cookie.

Nested Example #2 a. Calculate the probability that a cookie selected at random contains exactly 10 chocolate chips. b. Calculate the probability that in 17 randomly selected cookies at least 3 have exactly 10 chocolate chips in them.

Nested Example #3a An urn contains 6 red balls, 6 green balls, and 3 purple balls. You randomly reach in and pull out 4 balls, one at a time with replacement. For each part, in addition to answering the question, also state the distribution and parameters you are using:

Nested Example #3b What is the probability that you draw at least 2 purple balls? If you draw all 4 balls without replacement, what is the probability that you draw at least 2 purple balls?

Nested Example #3c Now you are dealing with a huge urn with 6000 red balls, 6000 green balls but only 3 purple balls. You return to the one-at-a-time with replacement method of drawing 400 balls. What is the probability that you draw at least 2 purple balls?

Nested Example #4 It rains 3 days per month on average in California. For each part below, in addition to answering the question, also state the distribution and parameters you are using

Nested Example #4 (cont) a) What is the probability that there are no rainy days next month? b) What is the probability that there will be 4 rainless months during the next year? c) What is the probability that there will be no rain in the next 3 months?

Nested Example #5a For the following 3 problems, assume each day (class period) is independent of all others. Your Stat 225 instructor says the phrase “Probability is AMAZING” an average of 3 times per class period (50 min). To amuse yourself, for the next 2 weeks (6 class periods), you count how many times your TA says “Probability is AMAZING” per class. What is the probability at least 2 days your TA says “Probability is AMAZING” exactly 5 times during a class period.

Nested Example #5b For the following 3 problems, assume each day (class period) is independent of all others. Your Stat 225 instructor also likes class participation. Every class your TA will randomly selects 5 students to participate for that class. Your class consists of 40 students, 25 of which are male. You record for the next 2 weeks (6 class periods) the number of male students your TA chooses each day. What is the probability that over the 4 days your TA chooses at least one male student to participate?

Nested Example #5c For the following 3 problems, assume each day (class period) is independent of all others. In your Stat 225 class, the probability a person falls asleep in any given class is 0.05 (all students fall asleep in class independent of any other students). Your class has 40 students total. For the next two weeks (6 class periods) you count how many students fall asleep during the class. What is the probability that at least one of the 6 days have more than 3 people that fall asleep in class?