In your third year at Warwick you often have a drink with a friend who enjoys probability (yes, you make some strange friends here). On the Sunday evening before the start of Term 2 she challenges you to a curious game of chance. Hiding a fair six-sided die (numbered 0,1,2,3,4,5) from your observation, she throws it ten times and secretly notes each outcome on a convenient beer mat. Then she turns to you and announces:
"I shall now read you the running averages of the dice throws, but in reverse order.''
(You must have looked puzzled, because she breaks off to explain)
"Let Xn be the result of the nth throw, so the nth running average is Yn=(X1+...+Xn)/n. I will read out the numbers Y10, Y9, ..., Y1 in that order.''
Clearly a flicker of comprehension must have somehow crossed your face, because she now continues:
"At any stage you can stop me reciting the sequence. If the last average I read out was Yn then with probability Yn/5 (arranged using that book of random numbers which I carry everywhere) I will buy you a pint and the game ends. If you don't stop me at all then clearly you get a pint with probability Y1/5. Otherwise you buy me a pint.''
You calculate rapidly: E[Y1/5]=E[X1/5]=(1/6) (0+1+...+5)/5=1/2 even if you wait till the last number, and you get to choose when to stop. Must be to your advantage!
"Seems a reasonably good deal to me,'' you say.
"I'm glad you think so;'' she replies, "before we begin you can buy me a pint to compensate. Then we can play the game K times, where K is as large as you think it should be so that on average you come out ahead.''
How big should K be?
The very next day you hear about this course, ST318 Probability Theory; by the end of the course you'll have learned enough to be able to answer the above question instantly without any calculation at all. All this and 15 CATS credit too. Now that is what you call fair! How can you resist?