MOXTMOET. 109 



1 / *720 SIP) ^ 



■which rule Peter adopts we should take ^ f --^ — p^ + --.-^j > that 



is, 1^ — ^t; as Paul's chance in Case 11. Thus in Case II. Paul's 

 51 . oO 



chance is less than in Case I. ; and therefore he should adopt the 

 rule of changing when he has a seven. This is one of the argu- 

 ments on which Nicolas Bernoulli relies. 



On the other hand his opponents, in effect, deny the correctness 

 of estimating it as an even chance that Peter will adopt either 

 of the two rules which have been stated. 



We have now to estimate the following chance. Peter has an 

 eight and Paul has not compelled him to change ; what is Peter's 

 chance ? Peter must argue thus : 



I. Suppose Paul's rule is to change a seven; then he now 

 has an eight or a higher card. That is, he must have one out of a 

 certain 23 cards. 



(1) If I retain my eight my chance of beating him arises only 

 from the hypothesis that his card is one of the 3 eights; that is, my 



chance is ^ . 



(2) If I change my eight my chance arises from the five h}^o- 

 theses that Paul has Queen, Knave, ten, nine, or eight; so that my 

 chance is 



23 ■ 50 "^ 23 ■ 50 "^ 23 ■ 50 "^ 23 ' 50 "^ 23 ' 50 ' 



210 



that is 



23 . 50 



II. Suppose Paul's rule is to retain a seven. Then, as before, 



7 



(1) If I retain my eight my chance is ^ . 



(2) If I change my eight my chance is 



4 3 4 7 4 11 4 15 3 22 4 26 



314 



27 ' 50 "^ 27 * 50 "^ 27 ■ 50 "^ 27 ' 50 "^ 27 ' 50 "^ 27 ■ 50 ' 



that is 



, 27 . 50 



