Set Theorems and Axioms of Probability

Set Theorems and Axioms of Probability - Set Theorems and...

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Unformatted text preview: Set Theorems and Axioms of Probability Random Experiment action whose outcome cannot be predicted with certainty beforehand. Sample Space set of all possible outcomes for a random experiment. This is denoted by . The possible outcomes themselves are denoted by (lowercase omega). An outcome is also known as a simple event. It is 1 specific thing that can happen in the random experiment. Examples of random experiments: 1 coin flip, 10 coin flips, 30 rolls of a die, time until the next customer shows up, time until the next success. Event is a subset of the sample space. (This can be an outcome or multiple outcomes) An event occurs if and only if the outcome of the random experiment is an element of the event. Examples: 1st toss is a heads, the 2 rolls sum to 7, within the first 15 minutes, on the 5th try. A B if and only if event B occurs every time event A occurs. USEFUL NOTATION: \ is used for "minus" or except. Example: represents the outcomes of 2 rolls of a die. Say A is the event that both rolls are a 5. Then we can write A complement as AC = \ {(5,5)} . Let us consider a random experiment that consists of tossing a coin three times. ={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}. Let A=event exactly (only) 2 tails B=event the first 2 tosses are tails C=event all 3 tosses are tails 1) 2) 3) 4) B C = A B = A C = A C = 5) ( A C )C = Mutually Exclusive Events Events A and B are said to be mutually exclusive if they cannot both occur when the random experiment is performed. In terms of sets this means that A and B are disjoint ( A B = ). Pairwise Mutually Exclusive For events A1 , A 2 , ... , they are called pairwise mutually exclusive if no 2 of them can occur when the random experiment is performed. In terms of sets this means that A1 , A 2 , ... are pairwise disjoint-- Ai A j = whenever i j. Note: In our coin example, A and C are mutually exclusive events. Also, corresponds to an event that cannot occur. We call such an event an impossible event. Say A={spaghetti, lasagna, baked ziti, breadsticks} and B={taco, tamale, enchilada, churro}. A and B are mutually exclusive events. They cannot both occur at the same time. (With this setup think about ordering at an Italian restaurant vs. a Mexican restaurant) If we added C={hot dog, cheeseburger, bratwurst, French fries}, then the events A, B, and C are pairwise mutually exclusive since no pair of events have common elements. Axioms of Probability Kolmogorov Axioms for a Probability Measure Let be the sample space for a random experiment. A function P defined on the events of is called a probability measure if it satisfies the following three conditions: 1) Nonnegativity: P(E) 0 for each event E. 2) Certainty: P()=1. 3) Additivity: If A1 , A 2 , ... is a countable collection of mutually exclusive events, then P( U A ) = P( A ) i i n n P ( A B ) = P ( A) + P ( B ) - P ( A B ) . If the events A and B are mutually exclusive, then P( A B) = P ( A) + P ( B) . Say event E is composed of several outcomes, then the probability of E is just the sum of the probabilities of those outcomes. Legitimate Probability Model A probability model is legitimate if it satisfies 1) and 2) from the Kolmogorov axioms. Probability must be mathematically valid and reflect the underlying nature of the random experiment under consideration (practical importance and application). Probability model is a mathematical description of the random experiment based on certain primary aspects or assumptions. Equallikelihood model all possible outcomes are equally likely to occur. Under this model we can think of probabilities as percentages (use the frequentist definition of probability). Most of the time, the equallikelihood model is referred to as classical probability. Under this model, P( E ) = n( E ) . n() Empirical Probability is based on the frequentist interpretation of probability or the law of large numbers. It relies on repeating the random experiment a large number of times and using the proportions of occurrence of the various outcomes or events as their respective probabilities. Properties of Probability P() = 0 Domination Principle If A B then P(A) P(B). Complementation Rule (Extremely important) Brief proof in class P(E)=1-P(EC ) Exhaustive Events A1 , A 2 , ... are said to be exhaustive events if at least one of them must occur when the random experiment is performed. In other words, the union is equivalent to the sample space. UA n n = Partition of a Sample Space Events A1 , A 2 , ... are said to form a partition of a sample space if they are both mutually exclusive and exhaustive. 1) 2) Ai A j = if i j UA n n = Law of Partitions Suppose A1 , A 2 , ... form a partition of the sample space. Then, for all events B, P(B)= P( A n n B) . What is the simplest partition? General Addition Rule (very important) P ( A B ) = P( A) + P( B) - P( A B ) InclusionExclusion Principle Basic case (3 events) P ( A B C ) = P ( A) + P ( B ) + P (C ) - P ( A B) - P( A C ) - P( B C ) + P( A B C ) Advanced case (more than 3 events) The idea is the same as above. The probability of the union of multiple events equals the sum of the individual probabilities, minus the sum of the probabilities of the events taken 2 at a time, plus the sum of the probabilities taken 3 at a time, minus the sum of the probabilities taken 4 at a time, etc. ...
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