ON DIRECT PROBABILITIES. 37 



P |5, 12? is [12,8] , PP/5, 12J is 12' 



c|4, 25 1- is [25, 22] divided by [1, 4] 



5. T there be an event composed of several others in 

 succession, of which the first may happen in, say 10 

 different ways, the second in 14, and the third (let there 

 be but three) in 6, then the compound event must be 

 one out of 10 x 14 X 6, which is the number of dif- 

 ferent ways in which it may happen. When all the 

 component events may happen in the same number of 

 ways, this reduces itself, both in principle and rule, to 

 the case of permutation with repetition. 



I now proceed to some problems requiring nothing 

 but a knowledge of the measure of probability and the 

 preceding rules. Any event of known circumstances 

 may be familiarly represented by a lottery of balls of 

 different colours. Thus, suppose ten Russian ships, 

 twelve French, and fourteen English, are expected in 

 port, or may have arrived. Let one be as likely to ar- 

 rive as another, and suppose it known that two have 

 arrived, but not of which nation they are. Let a cer- 

 tain advantage accrue to A, if they should happen to be 

 Russian and French, of which he is desirous of selling 

 his chance immediately. The question is precisely the 

 following : Let there be a lottery of green, white, and 

 red balls, 10, 12, and 14 in number. Let two have 

 been drawn, and suppose a certain advantage to arise to 

 A, if they be green and white. What should be given 

 to A, certain, in consideration of his chance ? 



Here we must first consider the number of ways in 

 which two can be drawn. All the balls are 36 in num- 

 ber, and the whole number of pairs is 36 X 35 -=- 2 

 or 630. Of these a green ball (1 out of 10) may be 

 paired with a white ball (1 out of 12) in 120 different 

 ways. Consequently, the probability of his gaining the 

 advantage is f g or - ? \. The probability against it is 

 therefore ~Hr : or out of every 2 1 possible events, 4 are 

 in ,favour of, and 17 against, the advantage. It is there- 

 D S 



