MA 2631 Probability Theory Assignment 4

MA 2631
Probability Theory
Assignment 4
due on Tuesday, September 14
based on Lectures of Chapter 3.1–3.2
1. We are given a die and six fair coins. First we roll the die and then we flip exactly the
number of coins the die shows. What is the probability that we get exactly two “heads”?
2. A microchip for a cellphone is produced by the factories A, B and C. Factory A
produces 30% of all microchips, B 50% and C 20%. A microchip produced by A is
defective in 2% of all cases, a chip produced in B is defective in 3% and a chip produced
in C is defective in 0.5% of all cases. Assume you get a cell phone with a defective
microchip. How likely is it that it was produced in factory A?
3. a) Let A, B, C be three events in a sample space Ω with a probability P satisfying
P[C] > 0. Show that
P[A ∪ B | C] = P[A | C] + P[B | C] − P[A ∩ B | C].
b) Let A1, . . . , An be mutually exclusive events in a sample space Ω and C an event in
Ω with P[C] > 0. Show that
P
h[n
i=1
Ai
| C
i
=
Xn
i=1
P[Ai
| C].
4. An AIDS test can detect an HIV infection with 99% accuracy, whereas it can accurately
identify the absence of an infection with 98% probability. Noting that approximately
0.6% of the US population is infected by HIV, how likely is it that a positive AIDS test
indicates indeed a HIV infection?
2
5. We are given 5 coins,two of them have heads on both sides, one tail on both sides and
two are regular with heads on one side and tails on the other. We pick a coin at random
and throw it. What is the probability that
i) the bottom side shows heads while we can’t see the upper side?
ii) the bottom side shows also heads if the upper side shows heads?
6. (Continuation from Assignment 3, problem 2) A person picks 13 cards out of a standard
deck of 52. Assume now that while picking one card is visible and the player recognizes
the ace of hearts among the 13 cards.
iii) What is now the probability that he has at least one ace in his hand?
iv) What is now the probability that he has exactly one ace in his hand?
8 points per problems
Additional practice problems (purely voluntary - no points, no credit, no
grading):
Standard Carlton and Devore, Section 1.4: Exercises 50, 51, 56, 57, 62, 72, 74
Hard Ann and Clara want to meet at the cafeteria between 1 p.m. and 2 p.m. The first that
arrives waits for 20 minutes and, if the other one does not show up, leaves the cafeteria.
i) Assuming that they arrive independently at a random time point between 1 p.m.
and 2 p.m., what is the probability that they will actually meet?
ii) Given that they met, what is the probability that the first of them waited for more
than 10 minutes?
Extra The Monty Hall Problem: As WPI has a strong connection to goats, here a classical
brain teaser about the game show Let’s Make a Deal:
”Suppose you’re on a game show, and you’re given the choice of three doors: Behind
one door is a car; behind the others, goats. You pick a door, say No. 1, and the host,
who knows what’s behind the doors, opens another door, say No. 3, which has a goat.
He then says to you, ”Do you want to pick door No. 2?” Is it to your advantage to
switch your choice?”
While there are many heuristics for the solution, can you set up a rigorous
mathematical formulation that, using conditional probabilities, allows you to calculate
explicitly the probabilities for winning given you switch or keep your choice? Try to
spell out as explicitly as possible which modeling assumptions you make.
For more information see https://en.wikipedia.org/wiki/Monty Hall problem.