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... What is the probability, if we pick one woman at random, that her height will be some value X? For instance, between 68 and 70 inches P(68 < X < 70)? Because the woman is selected at random, X is a random variable. ...
... What is the probability, if we pick one woman at random, that her height will be some value X? For instance, between 68 and 70 inches P(68 < X < 70)? Because the woman is selected at random, X is a random variable. ...
4. Random Variables, Bernoulli, Binomial, Hypergeometric
... A bag has 3 green jelly beans and 7 red jelly beans. If you extract 2 jelly beans, what is the probability that the 2 of them are red? Now suppose that you draw 5 jelly beans out of the bag. What is the probability that 3 are red and 2 are green? This is an example of a Hypergeometric random variabl ...
... A bag has 3 green jelly beans and 7 red jelly beans. If you extract 2 jelly beans, what is the probability that the 2 of them are red? Now suppose that you draw 5 jelly beans out of the bag. What is the probability that 3 are red and 2 are green? This is an example of a Hypergeometric random variabl ...
CountableSets1
... where the pattern is given by the picture. It’s a lot of trouble to give a formula for the subscripts — way more trouble than it’s worth — but it’s obvious that every lattice point will eventually appear on the list. What about the set of rational numbers? Every rational number can be written in the ...
... where the pattern is given by the picture. It’s a lot of trouble to give a formula for the subscripts — way more trouble than it’s worth — but it’s obvious that every lattice point will eventually appear on the list. What about the set of rational numbers? Every rational number can be written in the ...
Math SCO G1 and G2
... becomes the denominator of your theoretical probability. For example, when flipping a coin, there are two possible outcomes of this event. You can flip “heads” or “tails”. There are two outcomes in total. (Heads and Tails). So the denominator of your theoretical probability will be 2. So the theor ...
... becomes the denominator of your theoretical probability. For example, when flipping a coin, there are two possible outcomes of this event. You can flip “heads” or “tails”. There are two outcomes in total. (Heads and Tails). So the denominator of your theoretical probability will be 2. So the theor ...
Lecture 3 Counting and Equally Likely Outcomes
... any six for her ticket. The winning number is then decided by randomly selecting six numbers from the forty-four. So the first number can be chosen in 44 ways, and the second number in 43 ways, making a total of 44 × 43 = 1892 ways of choosing the first two numbers. However, if a person is allowed ...
... any six for her ticket. The winning number is then decided by randomly selecting six numbers from the forty-four. So the first number can be chosen in 44 ways, and the second number in 43 ways, making a total of 44 × 43 = 1892 ways of choosing the first two numbers. However, if a person is allowed ...
Lecture 20 and 21 1 PCP theorems using Parallel Repetition 2 H
... 2CSPW for which all the constraints can be satisfied and those for which atmost fraction of constraints can be satisfied. This can be used to show the hardness of approximation of several NP-Hard problems. However, for other problems we need “stronger” PCP theorems which have much smaller ’s with ...
... 2CSPW for which all the constraints can be satisfied and those for which atmost fraction of constraints can be satisfied. This can be used to show the hardness of approximation of several NP-Hard problems. However, for other problems we need “stronger” PCP theorems which have much smaller ’s with ...
REMARKS ON FOUNDATIONS OF PROBABILITY
... We shall here be interested in those formal theories in which only relational constants occur. Variables of higher orders (e.g. running over sets or classes of sets) may be allowed, but in such cases we shall assume that in the formulas each such variable is bounded by means of a quantifier. Only in ...
... We shall here be interested in those formal theories in which only relational constants occur. Variables of higher orders (e.g. running over sets or classes of sets) may be allowed, but in such cases we shall assume that in the formulas each such variable is bounded by means of a quantifier. Only in ...
ON PROBABILITY MEASURES FOR DEDUCTIVE SYSTEMS I
... 1. The problem of defining probability measures not simply on the class of sentences of a language, but on the class of its deductive systems (relative to some configuration of logical rules) was first raised by Mazurkiewicz [1]. In the paper – the second part of which seems never to have been publi ...
... 1. The problem of defining probability measures not simply on the class of sentences of a language, but on the class of its deductive systems (relative to some configuration of logical rules) was first raised by Mazurkiewicz [1]. In the paper – the second part of which seems never to have been publi ...
Midterm (Sample Version 2, with Solutions) Name:
... You choose ten of the eleven problems to complete. Indicate which problem you are skipping by marking the appropriate box in the upper-right hand corner of that page. If you do not mark any skip box, or if you mark more than one skip box, then whichever problem is skipped by most students becomes th ...
... You choose ten of the eleven problems to complete. Indicate which problem you are skipping by marking the appropriate box in the upper-right hand corner of that page. If you do not mark any skip box, or if you mark more than one skip box, then whichever problem is skipped by most students becomes th ...
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.