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Lecture #10: Continuity of Probability
... Lecture #10: Continuity of Probability Recall that last class we proved the following theorem. Theorem 10.1. Consider the real numbers R with the Borel σ-algebra B, and let P be a probability on (R, B). The function F : R → [0, 1] defined by F (x) = P {(−∞, x]}, x ∈ R, characterizes P. The function ...
... Lecture #10: Continuity of Probability Recall that last class we proved the following theorem. Theorem 10.1. Consider the real numbers R with the Borel σ-algebra B, and let P be a probability on (R, B). The function F : R → [0, 1] defined by F (x) = P {(−∞, x]}, x ∈ R, characterizes P. The function ...
Statistics Review Day 16: Counting, Probability, and Logic Problems
... four-digit pin numbers are possible if no number can be used twice and the first and third digits must be odd, and the second and fourth numbers must be even? ...
... four-digit pin numbers are possible if no number can be used twice and the first and third digits must be odd, and the second and fourth numbers must be even? ...
2006/07 late summer resit paper - Queen Mary University of London
... a) State and prove Baye’s theorem. I have two coins in my pocket. One of them is a fair coin and the other has probability 1/4 of coming up heads when tossed. b) I take a coin out of my pocket at random and toss it. What is the probability that the coin comes up heads? c) I take a coin out of my poc ...
... a) State and prove Baye’s theorem. I have two coins in my pocket. One of them is a fair coin and the other has probability 1/4 of coming up heads when tossed. b) I take a coin out of my pocket at random and toss it. What is the probability that the coin comes up heads? c) I take a coin out of my poc ...
5.1
... between _________ that describes the proportion of times the outcome would occur in a very _____ series of repetitions. The Practice of Statistics, 5 ...
... between _________ that describes the proportion of times the outcome would occur in a very _____ series of repetitions. The Practice of Statistics, 5 ...
Question 3 - Week of August 8
... create a 6-digit number. Find the probabilities of the following events. (a) The resulting number is divisible by 2. (b) The digits 2 and 3 appear consecutively in order (i.e., 23 appears in the number). (c) The digits 2 and 3 appear in order but not consecutively (i.e. 2 before 3, but at least one ...
... create a 6-digit number. Find the probabilities of the following events. (a) The resulting number is divisible by 2. (b) The digits 2 and 3 appear consecutively in order (i.e., 23 appears in the number). (c) The digits 2 and 3 appear in order but not consecutively (i.e. 2 before 3, but at least one ...
Problem Sheet 6
... (b) By expressing Y as a sum of m independent random variables, find its probability generating function. 3. Let X1 , X2 , . . . be a sequence of independent and identically distributed non-negative integer valued random variables, and let N be a non-negative integer valued random variable which is ...
... (b) By expressing Y as a sum of m independent random variables, find its probability generating function. 3. Let X1 , X2 , . . . be a sequence of independent and identically distributed non-negative integer valued random variables, and let N be a non-negative integer valued random variable which is ...
union
... in a room of 41 people is 90%. • To randomly select ___ birthdays, randInt (1, 365, __)L1:SortA(L1) This will sort the day in increasing order; scroll through the list to see duplicate birthdays. Repeat many times. • The following short program can be used to find the probability of at least 2 peop ...
... in a room of 41 people is 90%. • To randomly select ___ birthdays, randInt (1, 365, __)L1:SortA(L1) This will sort the day in increasing order; scroll through the list to see duplicate birthdays. Repeat many times. • The following short program can be used to find the probability of at least 2 peop ...
1 Introduction 2 Borel
... then provides us with an upper bound for each factor in the product P (∩k≥n Fkc ) ≤ ...
... then provides us with an upper bound for each factor in the product P (∩k≥n Fkc ) ≤ ...
6.3 Notes
... __)L1:SortA(L1) This will sort the day in increasing order; scroll through the list to see duplicate birthdays. Repeat many times. The following short program can be used to find the probability of at least 2 people in a group of n people having the same birthday : Prompt N : 1- (prod((seq((366-X ...
... __)L1:SortA(L1) This will sort the day in increasing order; scroll through the list to see duplicate birthdays. Repeat many times. The following short program can be used to find the probability of at least 2 people in a group of n people having the same birthday : Prompt N : 1- (prod((seq((366-X ...
Infinite monkey theorem
![](https://commons.wikimedia.org/wiki/Special:FilePath/Monkey-typing.jpg?width=300)
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.