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Lecture 2 - Equally likely outcomes

# Lecture 2 - Equally likely outcomes - ORIE 3500/5500...

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ORIE 3500/5500, Summer’11, Li Equally Likely Outcomes In many applications, it turns out that the sample space is finite and all the outcomes are “equally likely”, ie , there is no reason to believe that one of the outcomes has any special preference. Example: tossing a fair coin, rolling a fair die, choosing a card from a well-shuffled deck etc. [Classical Definition of Probability] Consider an experiment with a finite number of equally likely outcomes. Then for any event A , the probability of A is defined by P ( A ) = # possible outcomes in A # possible outcomes in . Example Suppose that Brian and Charles are gambling. Both of them roll a die and the person who has the higher number gets \$10 from the other. No transaction takes place in case of a tie. What is the probability that Brian wins the money? We can pose this problem as follows. 1) Specify the sample space and its possible outcomes. Let Ω = { ( i, j ) : 1 i, j 6 } . By direct counting, we find that there are 36 possible outcomes in Ω. 2) Specify the event and its possible outcomes. Let A be the event that Brian wins, i.e., A = { ( i, j ) , i > j } . By direct counting, we find that there are 15 possible outcomes in A . 3) Calculate the probability. P ( A ) = 15 / 36 = 5 / 12. In this example, it was easy to list the elements of the sample space Ω and the event A . More often than not, this is not the case. It becomes diﬃcult and cumbersome to list the elements of the sample space or any event. In many cases the counting principle comes to our aid.

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Lecture 2 - Equally likely outcomes - ORIE 3500/5500...

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