By Ronald Meester

Compactly written, yet however very readable, attractive to instinct, this creation to chance thought is a wonderful textbook for a one-semester direction for undergraduates in any course that makes use of probabilistic principles. Technical equipment is simply brought while precious. The path is rigorous yet doesn't use degree thought. The textual content is illustrated with many unique and astounding examples and difficulties taken from classical purposes like playing, geometry or graph idea, in addition to from purposes in biology, drugs, social sciences, activities, and coding idea. simply first-year calculus is needed.

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**Additional resources for A Natural Introduction to Probability Theory, Second Edition**

**Example text**

I am not sure whether this makes any sense, but it certainly expresses the idea that independence plays a crucial role in probability theory. We want two random variables to be called independent, if knowledge of the outcome of the ﬁrst has no eﬀect on the distribution of the second. Here is the formal deﬁnition. 1. The random variables X1 , X2 , . . , Xn are called independent if the events {X1 = x1 }, . . , {Xn = xn } are independent for any choice of x1 , . . , xn . The concept is best illustrated with some examples.

Note that in the solid line network, there is a connection from left to right, while there is no top to bottom connection in the dual network. 9. However, there is a better way to compute P (Ak ). Note that we have constructed the experiment in such a way that the events Bi are independent. Indeed, we built our probability measure in such a way that any outcome with k 1s and n − k 0s has probability pk (1 − p)n−k , which is the product of the individual probabilities. Hence we see that P (Ak ) = = P (B1 ∩ B2 ∩ · · · ∩ Bk−1 ∩ Bkc ) P (B1 )P (B2 ) · · · P (Bk−1 )P (Bkc ) = (1 − p)k−1 p.

A pack contains m cards, labelled 1, 2, . . , m. The cards are dealt out in a random order, one by one. Given that the label of the kth card dealt is the largest of the ﬁrst k cards, what is the probability that it is also the largest in the whole pack? 37 (de M´er´e’s paradox). Which of the following two events has the highest probability: (1) at least one 6, when we throw a die 4 times; (2) at least one double 6, when we throw two dice 24 times. 38. Let A1 , A2 , . . be events. Show that n P n Ai i=1 P (Ai ) − (n − 1), ≥ i=1 for all n = 1, 2, .