Download Homework 1 - UC Davis Statistics

Statistics 130A
F. J. Samaniego
Homework # 1
Due: Monday, June 27, 2011
From the text: Section 1.2, # 2, 4 6, 8, 12; Section 1.4, # 13, 14, 18; Section 1.5, # 8, 12;
Section 1.6, # 2, 8
Also, solve the following problems:
1. Prove the Bonferroni Inequality: For any n sets { Ai : i = 1,…,n},
n
P(
i 1
Ai' )  1 -
n

i 1
P( Ai ).
(Hint: Try to apply the “countable subadditivity” property of probabilities.)
2. The workers in a particular factory are 65% male, 70% married, and 45% married
male. If a worker is selected at random from this factory, find the probability that the
worker is (a) a married female, (b) a single female, (c) married or male or both.
3. Two cards are drawn at random (without replacement) from a 52 card deck. Find the
probability (a) that you draw a pair, and (b) that the first card drawn is larger than the
second. For concreteness, let an ace have value one.
4. Find conditions on a pair of sets A and B for which the following formula holds: P(A –
B) = P(A) - P(B); prove that this formula holds under these conditions.
5. You give a friend a letter to mail. He forgets to mail it with probability .2. Given that
he mails it, the Post Office delivers it with probability .9. Given that the letter was not
delivered, what’s the probability that it was not mailed?
6. Three bums take the late night bus from UCD to Davis Community Park. Each bum
chooses one of the three exits adjacent to the park randomly and independently. What is
the probability that the bums get off at three different exits?
7. Three CIA employees work independently at decoding an encrypted message. Their
respective probabilities of successfully decoding the message are 1/2, 1/3 and 1/4.
What’s the probability that the message gets decoded?
8. An urn contains 5 red marbles, 3 blue ones and 2 white ones. Three marbles are drawn
at random, without replacement. You got to the party late, but you overheard someone
say that the third marble drawn was not white. Given this fact, compute the conditional
probability that the first two marbles were of the same color.
Some practice problems for Stat 130A Discussion Session, Thursday, June 23.
1. Give a set-theoretic proof of the following statement: For any sets A, B1,…, Bn,
 n 
A   Bi  
 i 1 
n
( A  Bi ).
i 1
2. Some local researchers attribute the rampant bad breadth on the UCD campus to a new
brand of processed garlic being used at all campus dining establishments. It is known that
25% of the campus population has bad breadth, 10% chew tobacco and 5% have both
characteristics. If a campus citizen is chosen at random, find the probability that this
person either has bad breadth or chews tobacco, but not both.
3. Pak-Af Airlines services the cities of Islamabad and Kabul. On any given Tuesday, a
random passenger on this route is travelling on business (B) with probability .6, is
travelling for leisure (L) with probability .3 and is travelling for trouble-making (T) with
probability .1. Records show that the probability that Airport Security detains (D) a
traveler varies: P(D | B) = .2, P(D | L) = .3 and P(D | T) = .9. (a) What’s the probability
that a random traveler is detained? (b) Given that a particular traveler is detained, what
the probability he/she is traveling on business?
4. Three male members of the Davis Opera Society went to see La Boheme at the
Mondavi Center last night. They turned in their top hats at the hat check stand. When
they left, the hat check clerk was nowhere to be seen. Since all three hats looked the
same, they each took one of the hats at random. What’s the probability that at least one
of them got their own hat? (Hint: draw a tree.)
5. Contestants on a game show are asked to choose one of three doors to open. One door
has been randomly selected (each having probability 1/3) and a major prize has been
placed behind it. The other doors have no prize behind them. Monte knows where the
prize is. When contestants choose a door, Monte immediately offers them the opportunity
to change their minds. In doing so, he opens one of the two doors not chosen and shows
that there is no prize behind it. Contestants must then choose between sticking with the
original door chosen or switching to the other unopened door. Which strategy is best, or
are they actually equivalent? Give a probabilistic argument to support your answer. (Hint:
It is helpful to draw a tree, the first stage associated with the random placement of the
prize and the second stage associated with the (random) original choice of door 1, 2 or 3.)
6. Suppose two cards are to be drawn randomly from a standard 52-card deck. Compute
the probability that the second card is a spade when the draws are made a) with
replacement and b) without replacement. Compare these two answers and reflect on
whether the result you got works more generally (that is, with more than 2 draws). Now,
use your intuition to obtain the probability that the 52nd draw, without replacement, from
this 52-card deck will be a spade.