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Category Archives: Basic probability
Card matchings
Given two decks of \(n\) cards what is the probability that at least one of a pair of cards matches ?
Imagine the two decks of cards lined up, consider the event \(A_i\) to be that the \(i\)-th cards in the two decks match.
- Compute \[ \mathbf{P}(\bigcup_{i=1}^n A_i)\].
- (**)compute \[ \lim_{n\rightarrow \infty} ( \mathbf{P}( \bigcup_{i=1}^n A_i )\]
Monty Hall
Consider the Monty Hall problem:
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 [but the door is not opened], 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?
(You have observed that week after week on this show, the host always opens a door to revival a goat before giving you the option to switch)
- Without using conditional probabilities work out the probability of winning if you switch versus if you don’t.
- Compute the same probabilities using conditional probabilities.
- Now assume that the host is cruel. He only offers you a switch when you have chosen the car. (You figure this out by watching the show on TV before going.) Now should you switch when offered the chance ?
- Now lets assume that every morning the host gets up and with probability \(p\) acts like the original host and with probability \(1-p\) he acts like the cruel host from the previous question.
- Now what is the chance of wining if you switch ?
- Now what is the chance of wining if you don’t switch ?
- For which \(p\) should you switch and which \(p\) should you not switch ?
- Now let us assume that host just picks a door randomly after you have picked yours. If there is a car behind it the came is over. If there is a goat, he offers you the chance to switch. Should you switch when you have the chance ? why ?
Counting and geometry
1) How many ways can one order \(n\) people on a line ?
2) How many ways can one order \(n\) people on a circle or round table (note that rotations are considered equivalent )?
3) How many ways can one order \(n\) people on a circle or round table with invariance with respect to rotations and direction (clockwise versus counter-clockwise) ?
Stuffing Envelopes
You write a stack of thank you cards for people who gave you presents for your birthday. You address all of the envelopes but before you can stuff them you are called away. A friend tying to help you see the stack of cards and stuffs them in the envelops. Unfortunately they did not realize that each card was personalized and just stick them in the envelops randomly. Assuming there were \(n\) cards and \(n\) envelops, let \(X_{n}\) be the number of cards in the correct envelope.
- Find \(\mathbf{E} (X_n)\).
- Show that the variance of \(X_n\) is the same as \(\mathbf{E} (X_n)\).
- Is there any common distribution which has the above statistics ?
- (**) Show that
\[ \lim_{n\rightarrow \infty} \mathbf{P}(X_n =m) = \frac{1}{e\, m!}\]
Statespace of the NCAA Basketball tourniment (in the old days)
- Until 2000, the NCAA men’s Basketball tournament had 64 teams. They played each other in 6 rounds With the winner of each game moving one to play the winner of another game in a pre-specified order. If we assume the initial table of “who plays who” as well as who each winner plays is specified, give a concise description of the state space of all possible out comes. How many elements does the space have ?
- (*) If there are now \(2^n\) competitors and \(n\) rounds, answer the same questions as before.
[ Inspiration [GS2] p 1, # 3]
De Morgan’s Laws
Let \( A\), \(B\), and \(C_i\) be events.
- Show that \( (A \cup B )^c= A^c \cap B^c\).
- Show that \( (A \cap B)^c = A^c \cup B^c \).
- \[ \Big( \bigcup_{i=1}^n C_i \Big)^c = \bigcap_{i=1}^n C_i^c\]
- \[ \Big( \bigcap_{i=1}^n C_i \Big)^c = \bigcup_{i=1}^n C_i^c\]
- (**) Argue that the above previous two statements hold even when \(n=\infty\).
Is it True (Event Manipulation)?
Which of the following are identically true ? For those that are not, say when they are true.
- \(A \cup (B\cap C) = (A\cup B) \cap (A \cup C)\)
- \(A \cap (B\cap C)= (A \cap B) \cap C\)
- \( A \cup B) \cap C = A \cup (B\cup C) \)
- \( A \setminus (B\cap C) = (A\setminus B) \cup (A \setminus C)\)
[GS2, p.1 # 5]
Areas and circles
Let \(C_1\) be the circle of radius 2 centered at the origin \((0,0)\). Let \(C_2\) be the circle of radius 1 centered at the origin \((1,0)\).
- Draw a picture of matching the above description and shade the region inside of \(C_1\) but outside of \(C_2\).
- What is the area of the region you shaded ?
- What is the ratio of the shaded area to the area inside \(C_1\) ?
- What is the ratio of the area inside \(C_2\) to the area inside \(C_1\) ?
Algebra of events
Consider the sample space created by two coin flips. We will denote the outcome of first a head and then a tail by \((H,T)\). The even that there is exactly one head will be written \(\{(H,T), (T,H)\}\). Let \(\mathcal{A}\) denote the set of all events on this sample space.
- Write down all of the events in \(\mathcal{A}\). How many are there ?
- Pick a few elements of \(\mathcal{A}\) and show that the rules of being an algebra are in fact satisfied. Namely, if \(A,B \in \mathcal{A}\) then \( A \cup B \in \mathcal{A}\) and \(A^c \in \mathcal{A}\). (Here \(A^c\) is the compliment of \(A\) which is simply every event which is not is \(A\).)
- Why did I not ask you to verify by hand this property for all of the sets in \(\mathcal{A}\) ? Hint: can you tell me a lower bound on the number of such conditions you would have to check ?
- (*) Using the definition of an algebra, show that the empty set \(\emptyset\) is in \(\mathcal{A}\).
- (*) Using the definition of an algebra, show that if \(A, B \in\mathcal{A}\) then so must be \(A \cap B\).
- (*) Argue that if \(A_1, A_2,\dots,A_k \in\mathcal{A}\) then so are
\[ A_1\cup A_2\cup\cdots\cup A_k=\bigcup_{j=1}^k A_j\]
\[ A_1\cap A_2\cap\cdots\cap A_k=\bigcap_{j=1}^k A_j\]
Estimating differences of independent draws
- Show that if \(X\) and \(Y\) are independent random variables, then
\[\mathrm{Var}(X- Y)=\mathrm{Var}(X+Y)\] - Let \(D_1\) and \(D_2\) represents two draws at random with replacement from a population, with \(\mathbf{E}D_1=10\) and \(\mathbf{SD}(D_1)_1=2\). Find a number \(c\) so that
\[\mathbf{P}(|D_1 -D_2| < c) \geq .99\]
From [Pittman p. 203, #15]
Putting expectations together
Suppose \(\mathbf{E}(X^2)=3\), \(\mathbf{E}(Y^2)=4\) and \(\mathbf{E}(XY)=2\). What is \(\mathbf{E}[(X+Y)^2]\) ?
Expection and dice rolls
A standard 6 sided die is rolled three times.
- What is the expected value of the first roll ?
- What is the expected values of the sum of the three rolls ?
- What is the expected number of twos appearing in the three rolls ?
- What is the expected number of sixes appearing in the three rolls ?
- What is the expected number of odd numbers ?
Based on [Pitman, p. 182 #3]
Indicatior functions and expectations
Let \(A\) and \(B\) be independent events and let \(\mathbf{1}_A\) and \(\mathbf{1}_B\) be the associated indicator functions. Answer the following questions in terms of \(\mathbf{P}(A)\) and \(\mathbf{P}(B)\).
- Describe the distribution of \( \mathbf{1}_A\).
- What is \(\mathbf{E} \mathbf{1}_A\) ?
- Describe the distribution of \((\mathbf{1}_A +\mathbf{1}_B)^2\).
- What is \(\mathbf{E}(\mathbf{1}_A +\mathbf{1}_B)^2 \) ?
Canididate: Confidence interval
A pollster wishes to knows the percentage \(p\) of people in a population who intend to vote for a particular candidate. How large must a random sample with replacement be in order to be at least 95% sure that the simple percentage is within one percentage point of \(p\) ?
Dice rolls: Explicit calculation of max/min
Let \(X_1\) and \(X_2\) be the number obtained on two rolls of a fair die. Let \(Y_1=\max(X_1,X_2)\) and \(Y_2=\min(X_1,X_2)\).
- Display the joint distribution tables for \( (X_1,X_2)\).
- Display the joint distribution tables for \( (Y_1,Y_2)\).
- Find the distribution of \(X_1X_2\).
Combination of [Pitman, p. 159 #4 and #5]
Blocks of Bernoulli Trials
In \(n+m\) independent Bernoulli \((p)\) trials, let \(S_n\) be the number of successes in the first \(n\) trials, \(T_n\) the number of successes in the last \(m\) trials.
- What is the distribution of \(S_n\) ? Why ?
- What is the distribution of \(T_m\) ? Why ?
- What is the distribution of \(S_n+T_m\) ? Why ?
- Are \(S_n\) and \(T_m\) independent ? Why ?
- Are \(S_n\) and \(T_{m+1}\) independent ? Why ?
- Are \(S_{n+1}\) and \(T_{m}\) independent ? Why ?
Based on [Pitman, p. 159, #10]
Balls in boxes
A box contains 1000 bulbs, of which 2 are black and the rest are white.
- Which of the following is mostlikely to occur in 1000 draws with replacement from the box ? Fewer than 2 black balls, Exactly two black balls, more than two black balls.
- If two series of 1000 draws are made at random from the box, what is, approximately, is the chance that they produce the same number of black balls ?
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] )
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 \( i \in \{1,\dots,n\}\).
