By R. Meester

"The e-book [is] a superb new introductory textual content on likelihood. The classical means of training likelihood is predicated on degree idea. during this ebook discrete and non-stop chance are studied with mathematical precision, in the realm of Riemann integration and never utilizing notions from degree theory…. various themes are mentioned, akin to: random walks, vulnerable legislation of enormous numbers, infinitely many repetitions, powerful legislation of huge numbers, branching tactics, susceptible convergence and [the] primary restrict theorem. the speculation is illustrated with many unique and extraordinary examples and problems." Zentralblatt Math

"Most textbooks designed for a one-year direction in mathematical information hide chance within the first few chapters as coaching for the records to return. This booklet in many ways resembles the 1st a part of such textbooks: it is all likelihood, no facts. however it does the likelihood extra absolutely than traditional, spending plenty of time on motivation, rationalization, and rigorous improvement of the mathematics…. The exposition is generally transparent and eloquent…. total, it is a five-star booklet on likelihood that may be used as a textbook or as a supplement." MAA online

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**Extra resources for A Natural Introduction to Probability Theory**

**Sample text**

Hence we only need to count the number of outcomes in which no two birthdays coincide. How many outcomes have no common birtday? Well, there is no restriction on the ﬁrst, but when we know the birthday of person 1, we have only 364 possibilities for the scond, etcetera. Hence we conclude that the probability of having no common birthday is equal to 365 · 364 · · · (365 − r + 1) . (365)r Now you can check that this probability is less than 1/2 for r = 23. The very surprising conclusion is that in a collection of 23 people, the probability that at least two of them have the same birthday is larger than 1/2.

Clearly, we want the outcomes of diﬀerent ﬂips to be independent. Hence the probability that the ﬁrst two ﬂips both result in 1, should have probability p2 . This reasoning leads to the conclusion that any outcome with k 1s and n − k 0s, should have probability pk (1 − p)n−k . Does this make sense? I mean, is P thus deﬁned indeed a probability measure? 9. What is the probability that the ﬁrst 1 appears at the kth ﬂip of the coin? 10. The event in question can be written as Ak = {ω ∈ Ω : ω1 = · · · = ωk−1 = 0, ωk = 1}.

Finally, we agree from now on that ∞ − ∞ is not deﬁned. 44 Chapter 2. 1. The expectation of a random variable X is given by xP (X = x), E(X) = x whenever this sum is well deﬁned. Why is this deﬁnition reasonable? At the beginning of this section, we said that the expectation refers to the average value taken by a random variable. we will now explain why this is the case with the above deﬁnition. Let x1 , . . , xk be the outcomes of k independent random variables with the same distribution as some random variable X, and let, for each m, km be the k number of xi ’s which take the value m.