MA 2631 Probability Theory Assignment 5

MA 2631
Probability Theory
Assignment 5
due on Friday, September 17
based on Lectures of Chapter 3.3–4.1
1. Show that if A and B are independent events on a sample space Ω, then also Ac and B
are independent.
2. Suppose that A, B and C are independent events on a sample space Ω with
P[A ∩ B] 6= 0. Prove that
P[A ∩ C | A ∩ B] = P[C].
3. Assume that in a family the birth of a boy and a girl is equally likely and that the
family has n ≥ 2 children. Are the events
A . . . There is at least one boy and at least one girl in the family
B . . . There is at most one girl in the family
independent?
Hint: Note that the answer depends on n and you might want to prove a general
statement, e.g., by induction.
2
4. Let A, B, C be independent events on a sample space Ω with P[A] = 1
2
, P[B] = 2
3
and
P[C] = 3
4
. Calculate
P

A ∪ (B ∩ C)

.
5. Consider the probability mass distribution P[Y = i] = c · 0.1
i on the non-negative
integers for some constant c.
a) Calculate c.
b) Calculate P[Y = 0] and P[Y > 2].
c) Calculate P[Y ≤ 5 | Y > 2].
6. Assume you are flipping a fair coin until head appears the 5-th time. Let Y denote the
number of tails that occur. Calculate the probability mass distribution of Y .
8 points per problems
Additional practice problems (purely voluntary - no points, no credit, no
grading):
Standard Carlton and Devore, Section 1.5: 84, 85, 87, 91, 92; Section 2.2: 11, 12, 17
Extra In the early days of the Covid-19 pandemic it was often reported that the chance of an
infection is only relatively likely if one stays close to an infected person for (more than)
15 minutes1
. A clever WPI student was pondering the idea if this would mean that if
they meets three friends for 5 minutes each, the chance of an infection would be lower
than meeting one friend for 15 minutes and thus the risk would be manageable. Are
they right? To answer the question you can assume that
i) the 15 minutes estimates comes from the idea that with increasing time spent, the
infection gets likelier, and the 15 minutes mark a threshold amount, a probability p.
ii) the probability of an infection is equally likely in each of the 15 minutes.
iii) that an eventual Covid-19 infection of the friends is independent.
Note: practically there are good arguments against assumptions ii) and iii). More about
this and more reasonable assumptions later in class.
1E.g., ”Based on our current knowledge, a close contact is someone who was within 6 feet of an infected
person for at least 15 minutes starting from 48 hours before illness onset until the time the patient is isolated.”
https://www.cdc.gov/coronavirus/2019-ncov/php/principles-contact-tracing.html