ch 1-Introduction to Probability - STAT 230 Introduction to...

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STAT 230: Introduction to Probability and Random Variables Summer 2009 1 Probability 1.1 Sample space and Events Definition 1. Random Experiment A random experiment is an experiment for which the outcomes cannot be deter- mined ahead of time. Definition 2. Sample Space The sample space Ω is the collection of all possible outcomes. Elements ω in Ω are called simple outcomes. Definition 3. Event An event is a subset of the sample space. We say that an event occurs if and only if the outcome of the random experiment is an element of the event. Example 1. For example, consider the experiment where you draw a random card from a standard deck. The sample space consists of the 52 cards and one event is A = { card drawn is red } . Example 2. Suppose the experiment consists of tossing a coin twice. The sample space is Ω = { ( H, H ) , ( H, T ) , ( T, H ) , ( T, T ) } . The outcomes in the event A = { first toss is Head } are { (H,H),(H,T) } . Example 3. Consider the experiment where a pair of 6-sided die are rolled. The sample space is Ω = ( i, j ); i = 1 .. 6 , j = 1 .. 6 . The outcomes in A = { the second roll is 3 } are { (i,3);i=1..6 } . Example 4. Suppose we roll a 6-sided die and toss a coin. For the events A = { the die is even } and B = { the coin is Head } , then: A = { (2,H),(2,T),(4,H),(4,T),(6,H),(6,T) } . B = { (1,H),(2,H),(3,H),(4,H),(5,H),(6,H) } . A’ = { (1,H),(1,T),(3,H),(3,T),(5,H),(5,T) } . B’ = { (1,T),(2,T),(3,T),(4,T),(5,T),(6,T) } . A B = { (2,H),(4,H),(6,H) } . 1
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A B 0 = { (2,T),(4,T),(6,T) } . Definition 4. Mutually Exclusive Events A 1 , A 2 , ..., A n are mutually exclusive events if they cannot occur at the same time i.e A 1 , A 2 , .., A n are disjoint sets: A i A j = for all i 6 = j. In the case n=2, two events A and B are said to be mutually exclusive if A and B are disjoint: A B = Definition 5. Exhaustive events A 1 , A 2 , ..., A n are exhaustive events if A 1 A 2 ... A n = Ω . Definition 6. Partition A 1 , A 2 , ..., A n form a partition of the sample space Ω if they are mutually exclu- sive and exhaustive events. Example 5. A partition for the experiment where a 6-sided die is rolled and a coin is tossed is: A 1 = { the coin is Head } = { (1,H),(2,H),(3,H),(4,H),(5,H),(6,H) } and A 2 = { the coin is Tail } = { (1,T),(2,T),(3,T),(4,T),(5,T),(6,T) } . proof: A 1 A 2 = and A 1 A 2 = Ω . 1.2 Probability Definition 7. Probability (Kolmogorov’s axioms) Let Ω be the sample space for a random experiment. A function P defined on the events of Ω is called a probability (or probability measure) if it satisfies the following three conditions: (a)- P ( A ) 0 for each event A. (b)- P (Ω) = 1 . (c)- IfA 1 , .., A n are mutually exclusive events then P ( A 1 A 2 ... A n ) = P ( A 1 ) + P ( A 2 ) + ... + P ( A n ) for all n. Theorem 1. (a)- Complementation Rule : for any event A, with complement A’; P ( A 0 ) = 1 - P ( A ) . (b)- Addition Rule : for any two events A and B; P ( A B ) = P ( A ) + P ( B ) - P ( A B ) . (c)- Inclusion-Exclusion Rule : for any three events A, B and C; P ( A B C ) = P ( A )+ P ( B )+ P ( C ) - P ( A B ) - P ( A C ) - P ( B C )+ P ( A B C ) . 2
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(d)- Law of Partitions : If A 1 , A 2 , ..., A n form a partition of the sample space Ω , then for any event B; P ( B ) = P ( B A 1 ) + P ( B A 2 ) + ... + P ( B A n ) .
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