---
title: "Chapter 2.3 Practice: Conditional Probability"
author: "Your name here"
output: pdf_document
---


# 1. A hat contains six slips of paper with the numbers 1 through 6 written on them. Two slips of paper are drawn from the hat without replacement and the sum of the numbers is computed.

## a. What is the probability that the sum of the numbers is exactly 10?



## b. What is the probability that the sum of the numbers is at least 10?


## c. What is the probability that the sum of the numbers is exactly 10, given that it is at least 10?



\newpage
# 2. Roll two dice, one white and one red. Consider these events:

* A: The sum is 7
* B: The white die is odd
* C: The red die has a larger number showing than the white
* D: The dice match (doubles)

## a. Which pair(s) of events are disjoint?


## b. Which pair(s) are independent?


## c. Which pair(s) are neither disjoint nor independent?

\newpage

# 3. Let $A$ and $B$ be events. Show that $P(A\cup B|B)=1$.

\newpage
# 4. Suppose a die is tossed three times. Let $A$ be the event *the first toss is a 5*. Let $B$ be the event *the first toss is the largest number rolled* (the largest can be a tie). Determine via simulation whether $A$ or $B$ are independent.

\newpage
# 5. Bob Ross was a painter with a PBS television show ``The Joy of Painting'' that ran for 11 years. One report showed that 91% of Bob's paintings contain a happy little tree, 85% contain two or more happy trees. What is the probability that he painted a second happy tree to be a friend for the first happy tree?
