Fishin’ time!
Stark’s Pond contains 10 trout and 5 bluegill fish. Kyle catches a random number of fish (call the number \(X\)), where \(X \sim \text{Unif}(\{1,\ldots,4\})\). Once caught, that fish is removed from the pond and cannot be caught again. Each new fish comes uniformly from the remaining fish.
(a) What is the chance that Kyle catches all trout?
(b) Suppose all the fish that Kyle caught were trout. Given this information, what is the probability that he caught exactly 5 fish?
[Author Mark Huber. Licensed under Creative Commons.]
Introduction to exponential random variables
Let \(\Omega = \{(x,y):0 \leq x,0 \leq y \leq \exp(-x/2)\}\).
(a) What is the area of \(\Omega\)?
(b) Suppose \(U = (U_1,U_2)\) is drawn uniformly from \(\Omega\). Find \(\mathbf{P}(U_1 \leq 2.3)\).
(c) Find \(\mathbf{P}(U_2 \geq 1)\).
(d) For \(a\) an arbitrary positive real number, find \(\mathbf{P}(U_1 \leq a)\).
[Author Mark Huber. Licensed under Creative Commons.]
Time to play some bridge!
A hand in bridge consists of thirteen cards dealt out from a well shuffled deck.
(a) What is the probability that the bridge hand contains exactly 5 hearts?
(b) What is the probability that the bridge hand contains exactly 5 hearts and 5 spades?
(c) What is the probability that the hand contains exactly 5 cards from at least one suit?
[Author Mark Huber. Licensed under Creative Commons.]
Human error is the most common kind
Permanent Memories has three employess who burn Blu-ray discs. Employee 1 has a 0.002 chance of making an error, employee 2 has a 0.001 chance of making an error, and employee 3 has a 0.004 chance of making an error. The employees burn roughly the same number of discs in a day.
(a) What is the probability that a randomly chosen disc has an error on it?
(b) Given that a disc has an error, what is the probability that employee 1 was the culprit?
(c) Given that a disc has an error and employee 3 was on vacation the day it was burned, what is the probability that employee 2 was the culprit?
[Author Mark Huber. Licensed under Creative Commons.]
Drug testing with Bayes’ Rule
A new drug for leukemia works 25% of the time in patients 55 and older, and 50% of the time in patients younger than 55. A test group consists of 17 patients 55 and older and 12 patients younger than 55.
(a) A patient is chosen uniformly at random from the test group, the drug is administered, and it is a success. What is the probability the patient was in the older group?
(b) A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all four patients?
[Author Mark Huber. Licensed under Creative Commons.]
Adding binomials with equal success probability
Suppose \(X \sim \text{Bin}(10,0.2)\) and \(Y \sim \text{Bin}(5,0.2).\)
(a) Say \(X_1,\ldots,X_{10}\) are iid with \(X = X_1 + \cdots + X_{10}\). What distribution for the \(X_i\) makes this statement true?
(b) Say \(Y_1,\ldots,Y_{5}\) are iid with \(Y = Y_1 + \cdots + Y_{5}\). What distribution for the \(Y_i\) makes this statement true?
(c) Write \(X + Y\) in terms of \(X_1,\ldots,X_{10}\) and \(Y_1,\ldots,Y_{5}\).
(d) What is the distribution of \(X + Y\)?
[Author Mark Huber. Licensed under Creative Commons.]
A simple mean calculation
Suppose that \(X \in \{1,2,3\}\) and \(Y = X+ 1\), and \(\mathbf{P}(X = 1) = 0.3, \ \mathbf{P}(X = 2) = 0.5,\ \mathbf{P}(X = 3) = 0.2.\)
(a) Find \(\mathbf{E}(X)\).
(b) Find \(\mathbf{E}(Y)\).
(c) Find \(\mathbf{E}(X + Y)\).
[Author Mark Huber. Licensed under Creative Commons.]
Approximating binomial probabilities with Stirling
Let \(X\) be a binomially distributed random variable with parameters \(n = 1950\) and \(p = 0.342\).
(a) Approximate \(\mathbf{P}(X = 700)\) using Stirling’s approximation to eight significant digits.
(b) Find \(\mathbf{P}(X = 700)\) exactly to eight significant digits using Wolfram Alpha.
[Author Mark Huber. Licensed under Creative Commons.]
Introduction to Geometric random variables
Consider flipping a coin that is either heads (H) or tails (T), each with probability 1/2. The coin is flipped over and over (independently) until a head comes up. The outcome space is
\[ \Omega = \{H,TH,TTH,TTTH,\ldots\}. \]
(a) What is \( \mathbf{P}(TTH)\)?
(b) What is the chance that the coin is flipped exactly \(i\) times?
(c) What is the chance that the coin is flipped more than twice?
(d) Repeat the previous three questions for a unfair coin which has probability \(p\) of getting Tails.
[Author Mark Huber. Licensed under Creative Commons]
Roulette
Isabella is playing American roulette, where there are 38 spaces on a wheel, and there is a ball that is equally likely to land in each space. She plays 5 times, and the spins of the wheel are independent. If it lands in any of the 18 red spaces Isabella wins $1, but otherwise she loses $1. After her 5 plays, what is the probability that she ends up with more money than when she started?
[Author Mark Huber. Licensed under Creative Commons]
