Chapter 2. Combinatorial Probability

# Chapter 2. Combinatorial Probability - § 2 Combinatorial...

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Unformatted text preview: § 2 Combinatorial probability § 2.1 Definitions 2.1.1 Definition. Consider an experiment with a finite number of different possible outcomes. The set Ω of all possible outcomes is called the sample space . 2.1.2 An event A is a subset of Ω. It occurs if and only if the outcome of the experiment belongs to the subset A . 2.1.3 Definition. Assume that each outcome in Ω has EQUAL chance of occurring. The probability of event A is defined by P ( A ) = no. of outcomes in A no. of outcomes in Ω . 2.1.4 Example. If we roll a dice, possible outcomes are Ω = { 1 , 2 , 3 , 4 , 5 , 6 } . That “an even score turns up” is the event A = { 2 , 4 , 6 } . The probability of getting an even score is P ( A ) = 3 6 = 1 2 . 2.1.5 In the above definition, it is important that all outcomes in Ω must be equally likely, such that for each ω ∈ Ω, P ( { ω } ) = 1 no. of outcomes in Ω . Counter-example. When we toss a coin, it can land on a head, a tail OR on its edge. We may define Ω = { H , T , Edge } . The above definition instructs that P ( { H } ) = P ( { T } ) = P ( { Edge } ) = 1 / 3 , which is clearly not sensible. § 2.2 Examples 2.2.1 If we roll 2 dice, what is the probability that the total is 6? Here Ω = { (1 , 1) , (1 , 2) ,..., (1 , 6) , (2 , 1) ,..., (6 , 6) } contains 6 × 6 = 36 outcomes. The event of interest is A = { (1 , 5) , (2 , 4) , (3 , 3) , (4 , 2) , (5 , 1) } , 8 which contains 5 outcomes. Thuswhich contains 5 outcomes....
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## This note was uploaded on 04/29/2010 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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Chapter 2. Combinatorial Probability - § 2 Combinatorial...

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