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Category Archives: Manipulating Events
Meeting in a Tournament
A tennis tournament is organized for \(2^n\) players where each round is single elimination with \(n\) rounds. Two players are chosen at random.
- What is the chance that they meet in the first round or second round ?
- What is the chance they meet in the final or semi-final ?
- What is the chance they do not meet at all ?
[Sudov and Kelbert, p4 problem 1.2]
High Card Wins
Bob and Alice each have a box containing 3 numbered cards. Bob’s box has cards numbered 2,5 and 9. Alice’s box has cards numbered 3,4 and 8. Notice that the average value of the card in each box is the same. If each draws a card uniformly from their box, find the probability Alice wins. What is the probability Bob wins ?
Taking classes
There are 18 students in a room. How many students are not majoring in math or science or computer science ?
7 of them are math majors
10 of them are science majors
10 of them are computer science majors
4 of them are math and cs majors
3 of them are science and math majors
5 of them are cs and science majors
1 of them is a math, cs and science major
Pathway enrichment
A list of \(100\) genes are known to be part of the oxidative phosphorylation pathway.
My friend a molecular biologist screened the activity of \(5000\) genes in both diabetics and normal individuals.
He/she found \(500\) genes that were more active in normal individuals than diabetics.
Of these genes \(60\) of them belong to the list of genes that are part of the oxidative phosphorylation pathway.
What is the probability of this even happening randomly ? What is the scientific question behind the probability problem ?
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 ?
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]
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
\[ A_1\cup A_2\cup\cdots\cup A_k=\bigcup_{j=1}^k A_j\]
\[ A_1\cap A_2\cap\cdots\cap A_k=\bigcap_{j=1}^k A_j\]
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])
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 \(\mathbf{P}(A \cup B) = \mathbf{P}(A) + \mathbf{P}(B) – \mathbf{P}(A \cap B)\).
