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The matching problem
There are \(n\) letters addressed to \(n\) eople at different addresses. The \(n\) addresses are typed on \(n\) envelopes. A disgruntled secretary shuffles the letters and puts them in the envelopes in random order, one letter per envelope.
- Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.]
- What is the probability approximately, for large \(n\) ?
[ For example, the needed inclusion-exclusion formula is given in Problem 12, p. 31 in Pitman]
You will also need to know the number of elements in the set
\[ \{ (i_1,i_2,\cdots, i_k) : 1 \leq i_1 < i_2 < \cdots < i_k\leq n\} \]
which is discussed here.
Card hands: court cards
In a hand of 13 cards drawn randomly from a pack of 53, find the chance of:
- no court cards (J,Q,K,A);
- at least one ace but no other court cards;
- at most one kind of court card.
[Pitman p. 128, # 6]
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]
Three cornered duel: part two
- On the count of three, each person displays either ONE fingers or TWO fingers.
- If all three display the same number of fingers, then they throw again.
- If they do not all pick the same number, then two will match each other while the third person will have his own number. The third person wins and gets the piece of pizza.
Assume that each person will choose ONE or TWO with equal probability and independently of the others.
- Let \(N\) be the number of shoots necessary to determine a winner. Find \(\mathbf{E}(N)\) and \(\mathbf{SD}(N)\). Hint: No in finite sums are necessary. Express \(N\) in terms of a geometric random variable
- Now suppose that Archimedes and Bernoulli decide to cheat and they make a pact with each other.Archimedes will pick ONE with probability \(2/3\) and Bernoulli will pick TWO with probability \(2/3\).
- Show that this does indeed improve their chances of winning.
- What are \(\mathbf{E}(N)\) and \(\mathbf{SD}(N)\) now?
- Having made this pact, Archimedes wants to cheat Bernoulli as well. He will consider choose ONE with a probability \(p\) which may be different than\(2/3\). (Continue to assume that Bernoulli picks ONE with probability \(1/3\) and Cauchy picks ONE with probability \(1/2\)).
- Write Archimedes’ probability of winning the slice of pizza in terms of p. What value maximizes his chances ?
- Now suppose that Archimedes doesn’t care about winning, he just wants this to be over with. Write the expected number of games necessary to have a winner, \(\mathbf{E}(N)\), in terms of \(p\). What value of \(p\) minimizes \(\mathbf{E}(N)\)?
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?
The three-cornered duel
An intense dispute about the Axiom of Choice has led three of our favorite mathematicians, Archimedes,
Bernoulli and Cauchy into a duel. The contestants are to take turns in alphabetical order.
- Archimedes fires first, then Bernoulli (if he is still alive), then Cauchy (if he is still alive) and then back to Archimedes (if he is still alive), and so on.
- If a shot is successful, it is assumed to be fatal. The shooting proceeds until there is only one mam left standing.
From experience we know that Archimedes is not a very good shot (he lived 2000 years before gunpowder after all!) He is only successful 30% of the time. Bernoulli however is an excellent shot. He hits his target 100% of the time. Cauchy hits his target 70% of the time. After contemplating his predicament for a full night, Archimedes suddenly exclaims “Eureka!” Assuming every character will act in his own self-interest, what did Archimedes realize he should do with his first shot?
Looking for a rare couple
A certain advertising firm needs to find a recently married couple who are both born on April 19th. They send the intern to the city hall in NYC to look through the records.
- If she looks through 50,000 records, what is the chance that she finds at least one such couple?
- If you were to approximate the above probability with a limit theorem what distribution would you use? Explain why.
- If she relaxes her search criteria to only look for couples with the same birthday, what is the chance she finds at least one such couple after looking through 50,000 records?
- If you were to approximate the above probability with a limit theorem what distribution would you use? Explain why.
- Before starting, the intern is interested in estimating how long this task might take. How many records must she consider to have a 80% chance of finding a couple matching the first search criteria? What about the second search criteria?
In the above, assume that the date people are born is uniform over the year and each person’s birthday is independent of each other person’s birthday. Also, assume that the day one is born does not influence one’s choice of spouse.
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.
Cards: Independence
A card is selected at random from a deck of 52 playing cards. If \(E\) is the event that the card is a King and \(F\) is the event that it is a heart. Show that \(E\) and \(F\) are independent events
Coin Flips: describe events
Consider the probability space
\[ \Omega = \{ HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}\]
as the outcome of three consecutive tosses of a coin. (We make the reasonable assumption that all outcomes are are equally likely.) The event
\[ \{ HHH,TTT\}\]
is the event that all three tosses have the same outcome. Give a similar verbal description to each of the events bellow:
- \(\{HHH,HHT,HTH,HTT\}\)
- \(\{HTH,HTT,TTT,TTH\}\)
- \(\{HTT,HTH,HHT,HHH\}\)
- \(\{HTH,THH,TTH\}\)
- \(\{THT,HTT,TTH\}\)
- \(\{TTT,TTH,THT,HTT\}\)
- \(\{HHT,HHH,TTH,TTT\}\)
[Pitman, p31, #5] (Assign 1 and 5 first).
Inclusion/Exclusion: practice 1
Write down the expression in set notation corresponding to each of the following events:
- the event occurs if exactly one of the the events \(A\) and \(B\) occurs.
- the event which occurs if none of the events \(A\), \(B\), or \(C\) occurs.
- the event which occurs if exactly one of the events \(A\), \(B\), or \(C\) occurs.
- the event which occurs if exactly two of the events \(A\), \(B\), or \(C\) occurs.
- the event which occurs if exactly three of the events \(A\), \(B\), or \(C\) occurs.
(From [Pitman p.30, #2] )
Calculus: Differentiation
Perform the following differentiation:
- \[\frac{d\ }{dx} \Big(x^4\Big) \]
- \[\frac{d\ }{dx} \Big(x^2 \exp(-x)\Big) \]
- \[\frac{d\ }{dx} \Big(\ln(x^2) \Big) \]
Calculus: Infinite Sums
Evaluate the following infinite sums:
- \[ \sum_{k=1}^\infty \big(\frac14\big)^k\]
- \[ \sum_{k=1}^\infty \frac{3^k}{k!}\]
Calculus: Areas
Draw a picture of the region of the \(xy\)-plane were both x and y are between 0 and 1 and \(y \geq x^2\). Find the area of this region
Calculus : Exponentials integrals
Do the following integrals:
- \[ \int_0^1 x^3 dx\]
- (*) Hint: integrate by parts. \[\int_0^\infty x \exp(-x) dx\]
- (**) \[\int_0^\infty \exp(-\frac12 x^2) dx\] Try looking at \[\Big(\int_0^\infty \exp(-\frac12 x^2) dx\Big)\Big(\int_0^\infty \exp(-\frac12 y^2) dy\Big)=\int_0^\infty \int_0^\infty \exp(-\frac12 x^2) \exp(-\frac12 y^2)dx dy\] and then changing to polar coordinates. How des this help you figure out the original question ?
Practice with inclusion, exclusion.
Events \(A\), \(B\), and \(C\) are defined on a probability space. Find expressions for the
following probabilities in terms of \(\mathbf{P}(A)\), \(\mathbf{P}(B)\), \(\mathbf{P}(C)\), \(\mathbf{P}(AB)\), \(\mathbf{P}(AC)\), \(\mathbf{P}(BC)\), and \(\mathbf{P}(ABC)\).
- The probability that exactly two of the \(A\), \(B\), \(C\) occur.
- The probability that exactly one of these events occurs.
- The probability that none of these events occur.
Here the notation \(AB\) is short for \(A \cap B\) which is the event “both \(A\) and \(B\)”
( [Pitman, p. 31, # 10])
n – Dice : Max and Min
Suppose that a die has \(n\) sides. Compute the probability that:
- the maximum of the two numbers rolled is less than or equal to 2;
- the maximum of the two numbers rolled is less than or equal to \( i \in \{1,\dots, n\} \);
- the maximum of the two numbers rolled is equal to \( i \in \{1,\dots,n\}\).
Dice Rolls – Max and Min
Suppose two 4-sided dice are rolled. Find the probabilities of the following events:
- the maximum of the two numbers rolled is less than or equal to 2;
- the maximum of the two numbers rolled is less than or equal to 3;
- the maximum of the two numbers rolled is equal to 3;
- Repeat part c for the maximum equal to 1, 2, and 4.
- If M is the maximum of the two numbers, then
\[\mathbf{P}(M = 1) + \mathbf{P}(M = 2) + \mathbf{P}(M = 3) + \mathbf{P}(M = 4) = 1 \]
check that your answers for 3) and 5) satisfy this relationship.
Random Letters
Suppose a word is picked at random from this sentence. Find:
- the outcome space for this random experiment
- the chance the word has at least 4 letters;
- the chance that the word contains at least 2 vowels (a,e,i,o,u)
- the chance that the word contains at least 4 letters and at least 2 vowels.
- What is the distribution of the length of the word picked ?
- What is the distribution of the number of vowels in the word ?
(Based on [Pitman p.9 #2 and p. 31 #6)
Unions and intersections of events
Let \(A\) and \(B\) be any two events. Define the new events \(C\), \(\hat A\), and \(\hat B\) by \(C=A\cap B\), \(\hat A=A \cap B^c\), and \(\hat B = B \cap A^c\) where \(A^c\) is the compliment of \(A\) and \(B^c\) is the compliment of \(B\).
- Argue that \(A \cup B = \hat A \cup \hat B \cup C\) and that all three sets are mutually disjoint. i.e. \(\hat A\cap C = \emptyset\), \(\hat B\cap C = \emptyset\), and \(\hat A\cap \hat B = \emptyset\).
- Show that \(\mathbf{P}(A)= \mathbf{P}(\hat A) + \mathbf{P}(C)\) and \(\mathbf{P}(B)= \mathbf{P}(\hat B) + \mathbf{P}(C)\) .
- Show that \(\mathbf{P}(A \cup B) = \mathbf{P}(A) + \mathbf{P}(B) – \mathbf{P}(A \cap B)\).
Standardized Random Variables
Consider a random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). Define a new random variable \(Y\) by
\[Y=\frac{X-\mu}{\sigma}\,.\]
- Show that \(Y\) has mean 0 and variance 1.
- Show that if \(a \) is some number \[\mathbf{P}( Y > a) = \mathbf{P}( X > \mu + a\sigma )\]
