By R.K. Getoor

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March 24, 2008 17:9 B-595 ch02 This page intentionally left blank March 24, 2008 17:9 B-595 ch03 Chapter 3 A Day at the Races Do not trust the horse. 1. Kinds of Probability The kind of probability we have considered numerically thus far can be regarded as logical probability, which is to say that we can estimate probability by an exercise of logic, or reason. Usually this is based on the concept of symmetry; a coin has two sides neither of which is special in a probabilistic sense so each has an associated probability of 1/2.

Let us take an example of odds set by a bookmaker that would obviously be foolish. Consider a two-horse race where the bookmaker sets odds of 2/1 on each horse. A punter would not take long to work out that if he staked £1,000 on each horse his total stake on the race would be £2,000 but, whichever horse won, his return from the race would be £3,000. This case is easy to see without analyzing it in detail but more subtle examples of bad odds-setting can occur. Let us take a hypothetical race with 10 runners and the bookmaker, assessing the relative merits of the various runners, offers the following odds on a race for ﬁllies: Diana Dawn Lady Fairy Princess Olive Green Mayﬂy Dawn Chorus Missy Lovelorn Helen of Troy Piece of Cake 3/1 6/1 6/1 10/1 10/1 15/1 15/1 20/1 25/1 30/1 Although it is not obvious from a casual inspection of the list of odds this bookmaker is heading for certain ruin and any punter worth his salt could make a proﬁt on this race.

What is the probability that the three balls are red, green, and blue? 4. A town decides to run a local lottery based on the same model as the national lottery. There are only 20 numbers from which to pick, 1 to 20, and only four numbers are picked. Four winning numbers are selected from a drum-full of numbered balls plus a bonus ball. The top prize is for four correct numbers, the second prize is for three correct numbers plus the bonus ball and the third prize is for three correct numbers. What are the probabilities of winning each of these prizes with a single entry in the lottery?