Lectures on Elementary Probability

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8 CHAPTER 1. COMBINATORICS injective functions f from I to M with this indicator function is n!. In other words, given a subset A of M with n elements, the number of injective functions from I to M with range A is n!. This theorem says that there is a simple relation between unordered samples without replacement and ordered samples without replacement. For every one of the former there are exactly n! of the latter. Theorem 1.5 Let M be a set with m elements. The total number of indicator functions χ on M with sum n (that is, the total number of subsets A of M having n elements) is given by the binomial coefficient ( ) m = (m) n (1.10) n n! Proof: The total number of injective functions from I to M is the product of the number of subsets that may be taken as the range times the number of injections with that given range. This is ( m n ) n! = (m) n . (1.11) Note that the total number of subsets is the sum over n of the number of subsets of size n. This gives the identity m∑ ( ) m = 2 m . (1.12) n n=0
Chapter 2 Probability Axioms 2.1 Logic and sets In probability there is a set called the sample space S. An element of the sample space is called an outcome of the experiment. An event is identified with a subset E of the sample space S. If the experimental outcome belongs to the subset, then the event is said to happen. We can combine events by set theory operations. Each such operation corresponds to a logical operation. Union corresponds to or: s is in E ∪ F if and only if s is in E or s is in F. Intersection corresponds to and: s is in E ∩ F if and only if s is in E and s is in F. Complement corresponds to not: s is in E c if and only if s is in S but is not in E. The whole sample space S is the sure event. It corresponds to true: s is in S is always true. The empty set is the impossible event. It corresponds to false: s is in ∅ is always false. Two events E, F are exclusive if E ∩ F = ∅. This means that s in E and s in F is always false. 2.2 Probability The probability rules correspond to the logical operations. A probability assignment P assigns to each event E a number P [E] with 0 ≤ P [E] ≤ 1. The or rule: If E, F are exclusive, then The and rule: If E, F are independent, then P [E ∪ F ] = P [E] + P [F ]. (2.1) P [E ∩ F ] = P [E]P [F ]. (2.2) 9
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Lectures on Elementary Probability

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