53 



80 

 the chance of drawing 4 black and 2 white = -- , and the chance 



20 

 of drawing 2 black and 4 white = ~ . 



242. If n whole numbers taken at random be multiplied 

 together, the chance of the last digit in the product being 1, 3, 7, 



/\" 4"- 2" 



or 9 is f - ) ; of its being 2, 4, 6, or 8 is _, ; of its being 5 is 



5"-4" 10" -8" -5" + 4" 



-^r~; and of its being is - 1Q , . 



". If ten things be distributed among three persons, the 

 chance of a particular person having more than 5 of them is 

 1507 

 19683' 



244. If on a straight line of length a + b be measured at 

 random two lengths a, 6, the probability that the common part of 



c 

 these lengths shall not exceed c is -^, (c<a or b): and the 



probability of the smaller b lying entirely within the larger a is 

 a-b 



If on a straight line of length a+ b + c be measured at 

 random lengths a, 6, the chance of their having a common part 



shall not be greater than d is 7 ' ' .. . (</ < 



(c + a) (c + 6) 



a or b). 



346. There are m+j>+ q coins in a bag each of whir 1 

 equally lik.-ly to be a shilling or a sovereign; p + q being drawn, 

 p are shillings, and q sovereigns: prove that the value of th. 



expectation of the remaining sovereigns in the bag is - ., X. 



5, ;> = 2, 7^ 1, find the chance that if two more coins be 

 drawn tin -y will be a shilling and a sovereign, (1) when the coins 

 ;-u>ly drawn are not replaced, (2) when they are replaced. 



