Probability chances of winning

Question

The infamous 1984 World Chess Championship match between the reigning world champion Anatoly Karpov and 21-year-old Garry Kasparov was supposed to have been played until either player had won six games. Instead, it went on for 48 games: Karpov won five,

Kasparov won 3, and the other 40 games each ended in a draw. Alas, the match was controversially terminated without a winner.

We can deduce from the games Karpov and Kasparov played that, independently of other games, Karpov’s chances of winning each game were 5/48, Kasparov’s chances were 3/48, and the chances of a draw were 40/48. Had the match been allowed to continue indefinitely, what would have been Kasparov’s chances of eventually winning the match?

Answer 1

Hi Amy A.,

Sometimes these probability questions might seem to involve endless calculations -- but this one is just "ripe for the picking", as you will discover.

So what is the exact situation? Karpov has won 5 games, only needs 1 more game win for the championship. Kasparov has 3 wins, and would need 3 more, BEFORE Karpov gets any, to get the championship.

The first thing to do, is to disregard the non-essentials. Draw games are useless for determining a victor; therefore, consider only the universe in which the is a game winner to each of the next 3 games. Why only 3? B/c that's the minimum number required for Kasparov to win. You don't need to worry about what Karpov is doing meanwhile!

So, if the probabilities remained random, the ratio of Kasparov winning vs. Karpov winning is 3:5 or 3/8 as a probability. For 3 successive times that becomes: (You do the math!).

Of course, seldom are contests just random! Karpov had won games 3,6,7,9, and 27 (what does he favor, multiples of 3?) but Kasparov was on a roll with games 32, 47, and 48 (multiples of 2?); the Chess Federation terminated the match ("deep-sixed it", appropriately enough?) even though both players wanted to continue it, citing ostensible health concerns for the players. Really? With 48 games over 6 months?

Myself, I think the fix was in -- Kasparov went on to (narrow) victory in 1985, 1986, and 1987 subsequently.

--- Cheers, -- Mr. d.