Conditional probability
01
Conditional probability - algebra games
Assume that
, , and partition the sample space, and assume this data:
Find these values:
02
Conditioning relation
Suppose you know
and and . Calculate
and and .
03
Interpreting Multiplication - Fund performance
The odds of the Winning Fund outperforming the market in a random year are 15%. The odds that it outperforms the market in a 1-year period assuming it has done so in the prior year are 30%.
What is the probability of the Winning Fund outperforming the market in 2 consecutive years?
Bayes’ Theorem
04
Bayes’ Theorem - Stolen computer
Someone in a coffee shop “promises” to watch your computer while you’re in the bathroom.
If she does watch it, the probability that it gets stolen is 10%. If she doesn’t watch it, the probability that it gets stolen is 70%. You think there’s a 90% chance she is honest enough to watch it, having promised.
When you come back from the bathroom, the computer is gone. What is the probability that she witnessed the theft?
05
Bayes’ Theorem - Inferring die from roll
A bag contains one 4-sided die, one 6-sided die, and one 12-sided die. You draw a random die from the bag, roll it, and get a 4.
What is the probability that you drew the 6-sided die?
06
Bayes’ Theorem - DNA evidence
A crime is committed in a town of 100,000 citizens. After all 100,000 citizens’ DNA is analyzed, your friend Jim is found to have a DNA match to evidence at the scene. A forensics expert says that the probability of a random person matching this evidence is 0.01%. How likely is it that Jim is guilty?
Independence
07
Independence puzzle
Assume
, , and are mutually independent. Compute in terms of , , and .
08
Pairwise independent, not mutually independent: three coin flips
Flip a coin three times in sequence. Label events like this:
- exactly one heads among first and second flips - exactly one heads among second and third flips - exactly one heads among first and third flips Verify that
are pairwise independent but not actually mutually independent.
Tree diagrams
09
Bin of marbles
A bin contains 5 red marbles, 7 blue marbles, and 3 white marbles.
We draw a random marble. If it’s red, we put it back, if it’s not red, we keep it. We do this three times.
(a) What is the probability of getting red then white then blue?
(b) Suppose the last draw was blue. What is the probability that the first was red?
Counting
10
Counting outcomes - permutations and combinations
In a lottery, five distinct numbers are picked at random from
. How many possible outcomes are there: (a) If we care about the order of numbers.
(b) If the order does not matter.
11
Wisconsin flag 2 of 3 days
A kindergarten class hangs a random state flag (50 flags) on the wall every day. What is the probability that two days out of three given days have Wisconsin’s flag?
12
Drawing balls of distinct color
A bin contains 3 green and 4 yellow balls. Two balls are drawn out.
What is the probability that they are different colors?