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PROBABILITY. 333 Ex. 2. From a bag containing 5 red balls, 4 white balls, and 5 black balls, 6 balls are drawn at random. What is the chance that 3 are white, 2 black, and 1 red ?

The number of combinations of 4 white balls, taken 3 at a time, is 4.3.2 1.2.3 or 4. In the same manner the number of combinations of 5 black balls, taken 2 at a time, is 5.4 1.2 or 10. Since each of the 4 combinations of white balls may be taken with any one of the 10 combinations of black, and with each of the combinations so formed we may take any one of the 5 red balls, the total number of combinations will be 4.10.5 or 200. But the number of combinations of the entire number of balls, taken 6 at a time is 14.13.12.11.10.9 1.2.3.4.5.6 or 3003, hence the chance that 3 white, 2 black, and 1 red ball will be drawn at one time is 200 3003.

Ex. 3. A has 3 shares in a lottery in which there are 3 prizes and 6 blanks; B has 1 share in a lottery in which there is 1 prize and 2 blanks. Show that A's chance of success is to B's as 16 to 7.

A may draw 3 prizes in 1 way; he may draw 2 prizes and 1 blank in 3 2 1 2 6 ways; he may draw 1 prize and 2 blanks in 3 6.5 1.2 ways; the sum of these numbers is 64, which is the number of ways in which A can win a prize. Also he can draw 3 tickets in 9 8 7 1 2 3, or 84 ways; therefore A's chance of success = 64 84 = 16 21.

B's chance of success is clearly 1 3; therefore A's chance : B's chance = 16 21 : 1 3 = 16 : 7.

Or we might have reasoned thus: A will get all blanks in 6.5.4 1.2.3, or 20 ways; the chance of which is 20 84, or 5 21; therefore A's chance of success = 1 — 5 21 = 16 21. 405. From the examples given it will be seen that the solution of the easier kinds of questions in Probability requires nothing more than a knowledge of the definition of Probability, and the application of the laws of Permutations and Combinations.

EXAMPLES XXXVI.

1. A bag contains 5 white, 7 black, and 4 red balls; find the chance of drawing: (a) One white ball; (b) Two white balls; (r) Three white balls; (d) One ball of each color; (e) One white, two black, and three red balls.