Sanderson M. Smith

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PROBABILITY AND BRIDGE...VERY SIMPLE EXAMPLE

(Copy of note sent to students)Distinguished Cate history teacher and historian David Harbison produced a question that can be a learning experience for all of us.

Mr. Harbison's question related to the very sophisticated card came of BRIDGE. However, you don't have to know how to play BRIDGE in order to address the question. His question was basically this:

In a card game where you have a partner, suppose that you know that you and your partner have 9 of the 13 spades in the deck. In other words, your opponents have the 4 remaining spades distributed somehow in the two card hands they hold. You wonder how the spades are distributed in the opponents' hands. There is, of course, no way to know (since you can't look at their hands). But, good card players work with probabilities.In BRIDGE, you have what I will call a LEFT OPPONENT and a RIGHT OPPONENT. Here are theoretical probabilities relating to the distribution of the 4 spades.

LEFT OPPONENT

RIGHT OPPONENT

Probability 4 spades

0 spades

(_{4}C_{4})(.5)^4 = 0.0625 = 6.25%3 spades

1 spade

(_{4}C_{3})(.5)^4 = 0.25 = 25%2 spades

2 spades

(_{4}C_{2})(.5)^4 = 0.375 = 37.5%1 spade

3 spades

(_{4}C_{1})(.5)^4 = 0.25 = 25%0 spades

4 spades

(_{4}C_{0})(.5)^4 = 0.0625 = 6.25%

Note that the probabilities add up to 100%, as they should, since all possibilities for the distribution of four spades have been exhausted.I point out that these are theoretical probabilities based on randomness. In BRIDGE (an intellectual game) there is bidding before play begins. So, for example, the bidding of your LEFT OPPONENT might suggest that he/she has more spades than the RIGHT OPPONENT. The game of BRIDGE combines observation of human behavior with MATH POWER (theoretical probabilities)... as do many things in life.

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"Horse sense is what keeps horses from betting on what people will do."

-DAMON RUNYON

"Oh, many a shaft at random sent

Finds mark the archer little meant."

From THE LORD OF THE ISLES, by Sir Walter Scott.

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