Chris Morgan, MATH G160 csmorgan@purdue.edu February 15, 2011

Lecture 16

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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.

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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)

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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)

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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)

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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)

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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)

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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.

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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. 10

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:

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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?

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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?

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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

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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…