Statistics Chapter 5
... Find the following probabilities 1. Find the probability that the sum is a 2 2. Find the probability that the sum is a 3 3. Find the probability that the sum is a 4 4. Find the probability that the sum is a 5 5. Find the probability that the sum is a 6 6. Find the probability that the sum is a 7 7. ...
... Find the following probabilities 1. Find the probability that the sum is a 2 2. Find the probability that the sum is a 3 3. Find the probability that the sum is a 4 4. Find the probability that the sum is a 5 5. Find the probability that the sum is a 6 6. Find the probability that the sum is a 7 7. ...
Discrete Structures I - Faculty Personal Homepage
... 6. What is the probability that a poker hand contains a full house, that is, three of one kind and two of another kind? 7. What is the probability that the numbers 11,4, 17, 39, and 23 are drawn in that order from a bin containing 50 balls labeled with the numbers 1, 2, . . . , 50 if a) the ball sel ...
... 6. What is the probability that a poker hand contains a full house, that is, three of one kind and two of another kind? 7. What is the probability that the numbers 11,4, 17, 39, and 23 are drawn in that order from a bin containing 50 balls labeled with the numbers 1, 2, . . . , 50 if a) the ball sel ...
Chapter1
... A random experiment can have many possible outcomes; each outcome known as a sample point (a.k.a. elementary event) has some probability assigned. This assignment may be based on measured data or guestimates (“equally likely” is a convenient and often made assumption). Sample Space S : a set of all ...
... A random experiment can have many possible outcomes; each outcome known as a sample point (a.k.a. elementary event) has some probability assigned. This assignment may be based on measured data or guestimates (“equally likely” is a convenient and often made assumption). Sample Space S : a set of all ...
Chapter 4
... Probabilities for “at least” The multiplication rule can be used along with the complement rule to simplify problems involving “at least”. ...
... Probabilities for “at least” The multiplication rule can be used along with the complement rule to simplify problems involving “at least”. ...
A, B
... • A formal way to revise probabilities on the basis of new information is to use conditional probabilities. • Let A and B be any events with probabilities P(A) and P(B). Typically the probability P(A) is assessed without knowledge of whether B does or does not occur. However if we are told B has occ ...
... • A formal way to revise probabilities on the basis of new information is to use conditional probabilities. • Let A and B be any events with probabilities P(A) and P(B). Typically the probability P(A) is assessed without knowledge of whether B does or does not occur. However if we are told B has occ ...
Yr8-Probability (Slides)
... ? only provides a “sensible guess” for relative frequency/experimental probability the true probability of Heads, based on what we’ve observed. ...
... ? only provides a “sensible guess” for relative frequency/experimental probability the true probability of Heads, based on what we’ve observed. ...
eliminated
... tracking should make clear that the same kind of hands are possible either way, all equally likely. In both cases the shuffled deck determines the random sequence of dealing, with the same result. Another path involves just simple counting, without conditional probabilities. We can deal the cards in ...
... tracking should make clear that the same kind of hands are possible either way, all equally likely. In both cases the shuffled deck determines the random sequence of dealing, with the same result. Another path involves just simple counting, without conditional probabilities. We can deal the cards in ...
More generally, an unordered subset of k objects drawn a set
... = 6 outcomes on the first toss and n1 = 6 outcomes on the second toss, or n1n2 = 36. Now define the event A to correspond to exactly one 3 in 2 tosses. This can occur in two ways. First, we can obtain a 3 on the first toss (n1= 1, because there is only one way to achieve a 3 on the first toss) but n ...
... = 6 outcomes on the first toss and n1 = 6 outcomes on the second toss, or n1n2 = 36. Now define the event A to correspond to exactly one 3 in 2 tosses. This can occur in two ways. First, we can obtain a 3 on the first toss (n1= 1, because there is only one way to achieve a 3 on the first toss) but n ...
Section 5-2
... The probability of any event is between 0 and 1. All possible outcomes together must have probabilities whose sum is 1. If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula ...
... The probability of any event is between 0 and 1. All possible outcomes together must have probabilities whose sum is 1. If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula ...
Probability of Mutually Exclusive and Inclusive Events
... Mutually Inclusive Events Mutually inclusive events are events that can occur at the same time. For example, what is the probability of drawing either an ace or a spade randomly from a deck of cards? It’s possible for a card to be both an ace and a spade at the same time. When we consider the proba ...
... Mutually Inclusive Events Mutually inclusive events are events that can occur at the same time. For example, what is the probability of drawing either an ace or a spade randomly from a deck of cards? It’s possible for a card to be both an ace and a spade at the same time. When we consider the proba ...
Elementary probability examples, Counting techniques
... It then follows that we must define 0 ! = 1 . ...
... It then follows that we must define 0 ! = 1 . ...
Lecture 2 - Yannis Paschalidis
... If S has an infinite number of elements: Often, probability measure cannot then be defined in terms of atoms There may not exist a countable number of atoms Consider S = [0,1]. Atoms are {x}, where x ∈ [0,1] Note: there are an uncountable number of mutually disjoint atoms If all atoms are equally li ...
... If S has an infinite number of elements: Often, probability measure cannot then be defined in terms of atoms There may not exist a countable number of atoms Consider S = [0,1]. Atoms are {x}, where x ∈ [0,1] Note: there are an uncountable number of mutually disjoint atoms If all atoms are equally li ...
The probability of two events, A and B, are said to be independent
... question. In hw many different ways could you complete the test? a. 10 nPr 4= 5040 ways Permutation 16. If the order is not important, how many 7-card hands are possible? How many of these hands have exactly 6 cards of the same suit? a. 52 nCr 7 = 133784560 Combination b. 4 nCr 1 ∙ 13 nCr 6= 1716∙4= ...
... question. In hw many different ways could you complete the test? a. 10 nPr 4= 5040 ways Permutation 16. If the order is not important, how many 7-card hands are possible? How many of these hands have exactly 6 cards of the same suit? a. 52 nCr 7 = 133784560 Combination b. 4 nCr 1 ∙ 13 nCr 6= 1716∙4= ...
Chapter 5 - Elementary Probability Theory Historical Background
... Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling was Pierre Simon de Laplace, who is often credited with being the “father” of probability theory. In the twentieth century a coherent mathematical theory of probab ...
... Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling was Pierre Simon de Laplace, who is often credited with being the “father” of probability theory. In the twentieth century a coherent mathematical theory of probab ...
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.