BAYES' THEOREM 275 



1. At Smith's college an equal number of baseiiall and football games 



are played per season; 



2. Statistical records indicate that if a student participates in a base- 



ball game the probability is 2/100 that he will break an ankle 

 and that, likewise, the probability is 7/100 for the same con- 

 tingency in a football game. 



In view of the first of these two assumptions our conclusions as to the 

 cause of the accident may be based entirely on the information con- 

 tained in the second assumption. The odds are two to seven, so that 

 the a posteriori probabilities regarding the two admissible causes are: 



For baseball, 2/(2 + 7) = 2/9. 

 For football, 7/(2 + 7) = 7/9. 



Now consider this other example. A lone diner amused himself 

 between courses by spinning a coin. We elicited from the waiter that 

 in 15 spins, heads turned up seven times. Moreover, from our point 

 of observation, the size of the coin indicated that it was either a silver 

 quarter or a ten-dollar gold piece. What are the a posteriori proba- 

 bilities in favor of the silver quarter and the gold piece, respectively? 



If the lone diner were a professor from one of our eastern universities 

 we would not hesitate a moment in declaring that the coin spun was a 

 quarter. But it happens that the gentleman was a member of the 

 Cleveland Chamber of Commerce, dining at the Bankers' Club. We 

 must, therefore, give the matter more careful consideration. The 

 number of quarters and gold pieces usually carried by a banker and the 

 probabilities of obtaining the observed result by spinning coins are 

 relevant; let us assume, therefore, that: 



1. The small change purse of a Cleveland financier contains, on the 



average, ten-dollar gold pieces and quarters in the ratio of 

 eight to three. 



Moreover, we may assume (in fact we know) that: 



2. If either a quarter or a gold piece is spun 15 times, the probability 



that heads will turn up seven times is approximately 1/5. 



The second of these two items of information makes the a posteriori 

 probabilities depend entirely on the first item. Clearly the odds are 

 eight to three and we conclude; 



For a quarter, a posteriori probability = 3/ {3 + 8) = 3/11. 

 For a goldpiece, a posteriori probability = 8/(3 + 8) = 8/11. 



