50 ESSAY ON PROBABILITIES. 



of which there is an even chance ? If by H we here 

 mean a single H, the three events by which we propose 

 to attain it are not H, H, H, but H T T, T H T, and 

 TTH, the probability of each of which is 4, and, by the 

 application of the principle, ^ -f -^ + -^ or -^ is the 

 probability of the event, one single head. If by* 

 throwing a head we mean one head or more, the ways 

 under which it may be brought about are one head 

 only (involving the cases H T T, T H T, T T H) two 

 heads only (involving H H T, H T H, and T H H) 

 and three heads (involving H H H.) The probabilities 

 of these are -J , , and -1, or -J- is the chance required. 



Another error to which we are liable, is the wrong 

 estimation of the probability of the different cases. For 

 instance, a person is to go on until he throws H. and is to 

 win if he do it in less than five throws. There are then 

 five possible cases ; namely, H, T H, T T H, T T T H, 

 T T T T. In four of these cases he wins ; in the fifth he 

 loses. His chance of winning then appears to be 4. 

 But this supposes the five cases to be equally likely, 

 which is not true. Their several probabilities are \, , 

 , -jW and -^6 : not -$> -k> & c -> as supposed. Consequently 

 the sum of the probabilities which the several winning 

 cases actually have is -ff, or it is 15 to 1 that he wins. 

 If we put together all the cases, which four throws pre- 

 sent, thus, 



1. HHHH 5. HTHH 9. THHH 13. TTHH 



2. HHHT 6. HTHT 10. THHT 14. TTHT 



3. HHTH 7. HTTH 11. THTH 15. TTTH 



4. HHTT 8. HTTT 12. THTT 16. TTTT; 



all those which begin with H are 8 in number, those 

 which begin with T H are 4, with T T H, 2, and with 

 TTTH, one. Hence 8 -f 4 f 2 -f 1, or 15, is the 

 number of winning cases. But the argument against 

 us, is this : most of the preceding cases are impossible, 

 for the condition is, that the play shall stop as soon as 

 H occurs : so that, in fact, the only possible cases are, 

 H, T H, T T H, T T T H, T T T T. Let it be so ; 

 we must then represent the several events as follows : 



