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Independence of two hearts ?
Consider a deck of 52 cards. Let \(A\) be the event that the first card is a heart. Let \(B\) be the event that the 51st card is a heart.
What is \(\mathbf{P}(A)\) ? What is \(\mathbf{P}(B)\) ? Are \(A\) and \(B\) independent ?
Picking a box then a ball
Suppose that there are two boxes, labeled odd and even. The odd box contains three balls numbered 1,3,5 and the even box contains two balls labeled 2,4. One of the boxes is picked randomly by tossing a fair coin.
- What is the probability that a 3 is chosen ?
- What is the probability a number less than or equal to 2 is chosen ?
- The above procedure produces a distribution on \(\{1,2,3,4,5\}\) how does it compare to picking a number uniformly (with equal probability) ?
[Pitman p 37, example 5]
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.]
Airline Overbooking
An airline knows that over the long run, 90% of passengers who reserve seats for a flight show up. On a particular flight with 300 seats, the airline sold 324 reservations.
- Assuming that passengers show up independently of each other, what is the chance that the flight will be overbooked ?
- Suppose that people tend to travel in groups. Would that increase of decrease the probability of overbooking ? Explain your answer.
- Redo the the calculation in the first question assuming that passengers always travel in pairs. Are your answers to all three questions consistent ?
[Pitman p. 109, #9]
Drawing tickets
A box contains tickets marked \(1,2,…,n\). A ticket is drawn at random from the box.
Sampling with replacement — Then the ticket is replaced in the box and a second ticket is drawn at random. Find the probability of the following events:
a) the first ticket drawn is numer 1 and the second is number 2;
b) the numbers on the two tickets are consectutive integers;
c) the second number drawn is bigger than the first number.
Sampling without replacement — The ticket is not replaced in the box and a second ticket is drawn at random.
d) Repeat a)-c).
[Pitman page 9, Problem 3]
Coin Flips: typical behavior
A fair coin is tossed repeatedly. Considering the following two possible outcomes:
55 or more heads in the first 100 tosses.
220 or more heads in the first 400 tosses.
- Without calculations, say which of these outcomes is more likely. Why ?
- Confirm your answer to the previous question by a calculation.
[Pitman, p. 108 #3]
Leukemia Test
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 has 17 patients 55 and older and 12 patients younger than 55.
- A uniformly random patient is chosen from the test group, and the drug is administered and it is a success. What is the probability the patient was 55 and older?
- A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all of them?
School admissions
The ideal size of the freshman class of a small southern college is 150 students. Given that a student is admitted to the school, historical data indicates the student will actually attend with a probability 0.3. (We will assume that students make decisions independently of each other, even though this is certainly not true in reality). Approximately what is the chance that more than 150 students accept if 450 students are admitted.
