Homework #11 on Bayes's Theorem

Instructions: Using Bayes's Theorem, solve for conditional probabilities using the information given in the exercises. Make sure to:

1) Present your variable assignments, preferably using variables that intuitively correspond to the information given in the scenario. (As an example, I present the variable assignments I want you to use for Exercise #1.)

2) Present the formula for Bayes's Theorem using your variable assignments.

3) Feel free to proceed solving the problem in as step-by-step of a fashion, or not, as you see fit. You are welcome to proceed through it step-by-step, moving from a formula to a mixed expression - part formula, part equation - ultimately to an equation while doing the math along the way, or you can just go from formula to equation and do the math. Bottom line, I want to see all the variables replaced by figures before you just present your final answer.

4) Present your final answer in decimalized form and then as a percentage.

Exercise #1: There are two urns, each containing red and green balls, respectively. Urn #1 contains 80% red balls and 20% green balls. Urn #2 has 60% green balls and 40% red balls. You pick an urn at random, and then you draw balls from the urn in order to guess which urn it is. After each draw, the ball drawn is replaced. So, the probabilities remain consistent. What is the chance you picked from Urn #1 if you drew a red ball?

O = I picked Urn #1.

T = I picked Urn #2.

(It is up to you whether or not you want to use this variable and the figure that goes with it.)

R = I pulled a red ball.

Exercise #2: Shane found that in terms of computer ownership, his friends owned either a desktop or a laptop, but not both. He found that 60% of his friends own a desktop computer and 40% own a laptop computer. He also found out that 20% of the desktop-computer owners have Twitter accounts, but that 70% of the laptop owners have Twitter accounts. Given this information, figure out the probability that a friend who has a Twitter account is a laptop owner.

Exercise #3: There is a drug test for football players that has been confirmed to be 98% reliable if the player tested actually uses drugs. It has also been confirmed that if a player taking the test is not a drug user, the test is 93% reliable. Studies suggest that 20% of the players use drugs. So, for any given player who tested positive, what are the chances said player in fact uses drugs?

Exercise #4: Using the same scenario in Exercise #3, what are the chances a player is not a drug user given that said player tested negative?