---
title: "Ch 3.4 Practice: Named Discrete Distributions"
author: "Your name here"
output: pdf_document
---

> Questions in this homework are asked in English. Your first task is to define a random variable $X$ using an english sentence and writing the distributional notation. Practice writing out what the question is asking using symbols. Then find the answers using both theory (formulas) and simulation. 


# 1. Suppose you take a 20 question multiple choice test, where each question has four choices. You guess randomly on each question. You are interested in your overall score on the test. 

> Define the random variable and write the distributional notation here before part a of each question. (please delete this text)

## a. What is your expected score? 

## b. What is the probability you get 10 or more questions correct? 


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# 2. A recent national study showed that approximately 45\% of college students binge drink. Let $X$ equal the number of students in a random sample of size $n=12$ who binge drink. Using both the pmf by hand, and R commands, find the probability that

## a. $X$ is at most 2.

## b. $X$ is at least 1.

## c. Use simulation to obtain the mean and variance of $X$. 


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# 3. Stephen Curry is a 91% free throw shooter. He decides to shoot free throws until his first miss. 

## a. What is the probability that he shoots exactly 20 free throws (including the one he misses)?


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# 4. A couple decides to have children until a daughter is born. Assume the probability of a daughter is 0.5. What is the expected number of children of this couple? 


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# 5. Yruc Nepets is not nearly as good as Curry and only can make a free throw 19% of the time. 

## a. What is the expected number of free throws they will take until they make 5 baskets? 


## b. What is the variance in the number of free throws? 


## c. What is the probabilty that Yruc will make their 5th successful free throw in at most 10 shots? 
_Don't write out the equation here, use R functions and simulation only._

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# 6. There is a .85 probabilty that Sanjeev will show up to work exactly on time (not 1 minute late), independent of any other day. The company gives a $20 bonus for every employee that shows up to work exactly on time at least 5 days in a week. 

## a. Assuming the work week starts on Sunday (as day 1), what is the probability that Jose gets this bonus by thursday?



## b. What day on average will Sanjeev get this bonus?



## c. What is the probability that Sanjeev won't get this bonus?



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# 7. Flaws in a certain type of drapery material appear on the average of two in 150 square feet.  If we assume a Poisson distribution, find the probability of at most 2 flaws in 450 square feet.


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# 8. The Brown’s Ferry incident of 1975 focused national attention on the ever-present danger of fires breaking out in nuclear power plants.  The Nuclear Regulatory Commission has estimated that with present technology there will be, on average, one fire for every 10 years. Assume the incident of fires can be described by a Poisson distribution. 

## a. What is the expected value and variance of the number of fires over 10 years?


## b. Suppose that a certain state put three reactors on line in 2000. What is the probability that at least two fires will have occurred by 2015. _Careful! The time frame has changed. Be sure to adjust $\lambda$ accordingly$. 



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# 9. In the game of Scrabble, players make words using letter tiles. The tiles consist of 42 vowels and 58 non-vowels.

## a. If a player draws 7 tiles, what is the probability of getting 7 vowels?


## b. If a player draws 7 tiles, what is the probability of 2 or fewer vowels?



## c. What is the expected number of vowels drawn when drawing 7 tiles?



## d. What is the standard deviation of the number of vowels drawn when drawing 7 tiles?


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# 10. One of the popular tourist attractions in Alaska is watching black bears catch salmon swimming upstream to spawn. Not all "black" bears are black, some are tan-colored. Suppose that six black bears and three tan-colored bears are working the rapids of a salmon stream. Over the course of an hour, four different bears are sighted. 

What is the probability that there were more than twice as many black bears as tan-colored bears in those 4 sighted? 
