Probabilities Involving “and”, “or”, “not”
... liked to go fishing, and 12 said they don’t enjoy either activity. How many enjoy fishing but not sailing? ...
... liked to go fishing, and 12 said they don’t enjoy either activity. How many enjoy fishing but not sailing? ...
But Is it Random?
... Sequences generated by a computer are called “pseudo random” sequences, because a formula is applied to an inputted number called a “seed”. Each time a new number is generated, that number becomes the new seed. This means that identical sequences can be generated by using the same seed at the start ...
... Sequences generated by a computer are called “pseudo random” sequences, because a formula is applied to an inputted number called a “seed”. Each time a new number is generated, that number becomes the new seed. This means that identical sequences can be generated by using the same seed at the start ...
Lecture 2
... from a lottery, empirical observation) or s/he came up with them on her/his own. • The ultimate aim is to present a succinct way to capture rational human behavior when faced with situations of uncertainty. • This theory will not be perfect – we will point out many shortcomings. However, it is the b ...
... from a lottery, empirical observation) or s/he came up with them on her/his own. • The ultimate aim is to present a succinct way to capture rational human behavior when faced with situations of uncertainty. • This theory will not be perfect – we will point out many shortcomings. However, it is the b ...
Chapter5.1to5.2
... a discrete random variable X is one that can take on only a finite number of values for example, rolling a die can only produce numbers in the set {1,2,3,4,5,6} rolling 2 dice can produce only numbers in the set {2,3,4,5,6,7,8,9,10,11,12} choosing a card from a complete deck (ignoring suit) can prod ...
... a discrete random variable X is one that can take on only a finite number of values for example, rolling a die can only produce numbers in the set {1,2,3,4,5,6} rolling 2 dice can produce only numbers in the set {2,3,4,5,6,7,8,9,10,11,12} choosing a card from a complete deck (ignoring suit) can prod ...
In addition to the many formal applications of probability theory, the
... Definition 2.1 Two events are independent if P(AB) = P(A)P(B). Suppose that P(A) > 0 and P(B) > 0, then it follows from the definitions of independence and conditional probability that A and B are independent if and only if: P(A | B) = P(A) and P(B | A) = P(B), that is, learning that B (A) occurred ...
... Definition 2.1 Two events are independent if P(AB) = P(A)P(B). Suppose that P(A) > 0 and P(B) > 0, then it follows from the definitions of independence and conditional probability that A and B are independent if and only if: P(A | B) = P(A) and P(B | A) = P(B), that is, learning that B (A) occurred ...
Section 7B: Combining Probabilities
... Section 7B: Combining Probabilities Example. There are 10 freshmen and 15 sophomores in a class, and two must be selected for the school senate. If each person is equally likely to be selected, what is the probability that both people selected are ...
... Section 7B: Combining Probabilities Example. There are 10 freshmen and 15 sophomores in a class, and two must be selected for the school senate. If each person is equally likely to be selected, what is the probability that both people selected are ...
Lecture 1
... Probability theory has its roots in games of chance, such as coin tosses or throwing dice. By playing these games, one develops some probabilistic intuition. Such intuition guided the early development of probability theory, which is mostly concerned with experiments (such as tossing a coin or throw ...
... Probability theory has its roots in games of chance, such as coin tosses or throwing dice. By playing these games, one develops some probabilistic intuition. Such intuition guided the early development of probability theory, which is mostly concerned with experiments (such as tossing a coin or throw ...
Probability Theory
... 2.1 How many different five-card hands of poker include four cards of the same rank? 2.2 How many five-card hands of poker include both three of a kind (three cards of the same rank), and a pair (two cards of the same rank)? This hand is called a full house. 2.3 How many five-card hands of poker inc ...
... 2.1 How many different five-card hands of poker include four cards of the same rank? 2.2 How many five-card hands of poker include both three of a kind (three cards of the same rank), and a pair (two cards of the same rank)? This hand is called a full house. 2.3 How many five-card hands of poker inc ...
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Lecture # 3 Counting Techniques 1. Multiplication Principle: Let E
... Pick a no. Pick a no. Pick a no. for Dial 1 for Dial 2 for Dial 3 E1 E2 E3 Where E1 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => n(E1) = 10 E2 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => n(E2) = 10 E3 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => n(E3) = 10 This means, there are 10 choices for dial 1, 10 for dial 2 and 10 for ...
... Pick a no. Pick a no. Pick a no. for Dial 1 for Dial 2 for Dial 3 E1 E2 E3 Where E1 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => n(E1) = 10 E2 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => n(E2) = 10 E3 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} => n(E3) = 10 This means, there are 10 choices for dial 1, 10 for dial 2 and 10 for ...
Members of random closed sets - University of Hawaii Mathematics
... Let e be the probability of extinction for a GW-tree. By Lemma 2 we have e = pp , so since p > 1/2, e < 1. Thus there is a computable function (n, `) 7→ mn,` such that for all n and `, m = mn,` is so large that em ≤ 2−n 2−2` . Let Φ be a Turing reduction so that ΦG (n, `), if defined, is the least L ...
... Let e be the probability of extinction for a GW-tree. By Lemma 2 we have e = pp , so since p > 1/2, e < 1. Thus there is a computable function (n, `) 7→ mn,` such that for all n and `, m = mn,` is so large that em ≤ 2−n 2−2` . Let Φ be a Turing reduction so that ΦG (n, `), if defined, is the least L ...
Insufficient Reason Principle - Progetto e
... This principle can be included in the epistemic probabilities, where different interpretations can co-exist together. One of the best definition of the principle was introduced by Keynes in 1921, arguing that ‘if there is no known reason for predicating of our subject one rather than another of seve ...
... This principle can be included in the epistemic probabilities, where different interpretations can co-exist together. One of the best definition of the principle was introduced by Keynes in 1921, arguing that ‘if there is no known reason for predicating of our subject one rather than another of seve ...
Infinite monkey theorem
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.