Chapter 1 Discrete Probability Distributions - DIM

28 CHAPTER 1. DISCRETE PROBABILITY DISTRIBUTIONSby seeing what kind of a bet you would be willing to make. For example, supposethat you are willing to make a 1 dollar bet giving 2 to 1 odds that Dartmouth willwin. Then you are willing to pay 2 dollars if Dartmouth loses in return for receiving1 dollar if Dartmouth wins. This means that you think the appropriate probabilityfor Dartmouth winning is 2/3.Let us look more carefully at the relation between odds and probabilities. Supposethat we make a bet at r to 1 odds that an event E occurs. This means thatwe think that it is r times as likely that E will occur as that E will not occur. Ingeneral, r to s odds will be taken to mean the same thing as r/s to 1, i.e., the ratiobetween the two numbers is the only quantity of importance when stating odds.Nowifitisrtimes as likely that E will occur as that E will not occur, then theprobability that E occurs must be r/(r + 1), since we haveP (E) =rP(Ẽ)andP (E)+P(Ẽ)=1.In general, the statement that the odds are r to s in favor of an event E occurringis equivalent to the statement thatP (E) ==r/s(r/s)+1rr+s .If we let P (E) =p, then the above equation can easily be solved for r/s in terms ofp; we obtain r/s = p/(1 − p). We summarize the above discussion in the followingdefinition.Definition 1.4 If P (E) =p, the odds in favor of the event E occurring are r : s (rto s) where r/s = p/(1 − p). If r and s are given, then p can be found by using theequation p = r/(r + s).✷Example 1.12 (Example 1.9 continued) In Example 1.9 we assigned probability1/5 to the event that candidate C wins the race. Thus the odds in favor of Cwinning are 1/5 :4/5. These odds could equally well have been written as 1 : 4,2 : 8, and so forth. A bet that C wins is fair if we receive 4 dollars if C wins andpay 1 dollar if C loses.✷Infinite Sample SpacesIf a sample space has an infinite number of points, then the way that a distributionfunction is defined depends upon whether or not the sample space is countable. Asample space is countably infinite if the elements can be counted, i.e., can be putin one-to-one correspondence with the positive integers, and uncountably infinite