幻灯片 1
... then P Ai P ( Ai ) i 1 i 1 3) P ( AC ) 1 P ( A). 4) If B A then P ( A B ) P ( A) P ( B ) and P ( B ) P ( A) 5) P(A)≤1 for all events A. 6) If A and B are arbitrary events then P ( A B ) P ( A) P ( B ) P ( A B ) and hence P ( A B ) P ( A) P ( B ). ...
... then P Ai P ( Ai ) i 1 i 1 3) P ( AC ) 1 P ( A). 4) If B A then P ( A B ) P ( A) P ( B ) and P ( B ) P ( A) 5) P(A)≤1 for all events A. 6) If A and B are arbitrary events then P ( A B ) P ( A) P ( B ) P ( A B ) and hence P ( A B ) P ( A) P ( B ). ...
Discrete probability
... This generalises: if A1 , . . . , An are events that are disjoint from each other, then P (A1 ∪ . . . ∪ An ) = P (A1 ) + . . . + P (An ). A function P with domain the set of events, and satisfying (1) and (2), is called a probability function. ...
... This generalises: if A1 , . . . , An are events that are disjoint from each other, then P (A1 ∪ . . . ∪ An ) = P (A1 ) + . . . + P (An ). A function P with domain the set of events, and satisfying (1) and (2), is called a probability function. ...
Lecture_5 - New York University
... • Probability of an event is the ratio between the number of outcomes that satisfy the event to the total number of possible outcomes P(E) = N(E)/N(S) for event E and sample space S • Rolling a pair of dice and card deck as sample random processes ...
... • Probability of an event is the ratio between the number of outcomes that satisfy the event to the total number of possible outcomes P(E) = N(E)/N(S) for event E and sample space S • Rolling a pair of dice and card deck as sample random processes ...
Ch 5
... A Conditional Probability is the probability of a particular event occurring, given that another event has occurred. ...
... A Conditional Probability is the probability of a particular event occurring, given that another event has occurred. ...
Chapter 2 Discrete Random Variables
... We are sometimes interested in a summary of certain properties of a random variable. Ex: Instead of comparing your grade with each of the other grades in class, as a first approximation you could compare it with the class average. Ex: A fair die is thrown in a casino. If 1 or 2 shows, the casino wil ...
... We are sometimes interested in a summary of certain properties of a random variable. Ex: Instead of comparing your grade with each of the other grades in class, as a first approximation you could compare it with the class average. Ex: A fair die is thrown in a casino. If 1 or 2 shows, the casino wil ...
Slides for Chapter
... A Conditional Probability is the probability of a particular event occurring, given that another event has occurred. ...
... A Conditional Probability is the probability of a particular event occurring, given that another event has occurred. ...
A ∩ B - Cloudfront.net
... process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on. If A is any event, we write its probability as P(A). In the dice-rolling example, suppose we define event A as “sum is 5.” ...
... process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on. If A is any event, we write its probability as P(A). In the dice-rolling example, suppose we define event A as “sum is 5.” ...
Lecture 21 Approximation and Nested Problems
... number of Kings one player may have, and answer the following questions. 1. Is X a discrete or continuous random variable? 2. Find an appropriate probability distribution that ...
... number of Kings one player may have, and answer the following questions. 1. Is X a discrete or continuous random variable? 2. Find an appropriate probability distribution that ...
Probability Activity
... students what is the probability the next coin will be heads? Have students give explanations for their answers. Then pose this question to the students. "Suppose I flip all 10 coins, how many will land on heads and how many will land on tails?" Allow several students to give their answers and reaso ...
... students what is the probability the next coin will be heads? Have students give explanations for their answers. Then pose this question to the students. "Suppose I flip all 10 coins, how many will land on heads and how many will land on tails?" Allow several students to give their answers and reaso ...
Ars Conjectandi
Ars Conjectandi (Latin for The Art of Conjecturing) is a book on combinatorics and mathematical probability written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way, and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations—the aforementioned problems from the twelvefold way—as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work.