Then of the 8 triangles so formed it is evident that the circle will be escribed to 6 and inscribed in 2; and as this is true whatever be the original directions of P, Q, R, the required result follows.

*482. Questions in Probability may sometimes be conveniently solved by the aid of co-ordinate Geometry.

Example. On a rod of length a+b+c, lengths a, b are measured at random: find the probability that no point of the measured lines will coincide.

Let AB be the line, and suppose AP=x and PQ=a; also let a be measured from P towards B, so that x must be less than b+c. Again let AP'=y, P'Q'=b, and suppose P'Q' measured from P' towards B, then y must be less than a+c.

Now in favourable cases we must have AP'>AQ, or else AP>AQ',

Take a pair of rectangular axes and make OX equal to b+c, and OY equal to a + c.

Draw the line y=a+x, represented by TML in the figure; and the line x=b+y represented by KR.

Then YM, KX are each equal to c, OM, OT are each equal to a.

The conditions (1) are only satisfied by points in the triangles MYL and KXR, while the conditions (2) are satisfied by any points within the rectangle OX, OY;

c2

́(a+c) (b+c) •

*483. We shall close this chapter with some Miscellaneous Examples.

Example 1. A box is divided into m equal compartments into which n balls are thrown at random; find the probability that there will be p compartments each containing a balls, q compartments each containing b balls, compartments each containing c balls, and so on, where

Since each of the n balls can fall into any one of the m compartmen in the total number of cases which can occur is m", and these are all equally likely. To determine the number of favourable cases we must find the number of ways in which the n balls can be divided into p, q, r, parcels containing a, b, c, balls respectively.

...

First choose any s of the compartments, where s stands for p + q + r + ... ; the number of ways in which this can be done is

m

sm-s

.(1).

Next subdivide the s compartments into groups containing p, q, r,... severally; by Art. 147, the number of ways in which this can be done is

Lastly, distribute the n balls into the compartments, putting a into each of the group of p, then b into each of the group of q, c into each of the and so on. group of "", The number of ways in which this can be done is

Hence the number of ways in which the balls can be arranged to satisfy the required conditions is given by the product of the expressions (1), (2), (3). Therefore the required probability is

Example 2. A bag contains n balls; k drawings are made in succession, and the ball on each occasion is found to be white: find the chance that the next drawing will give a white ball; (i) when the balls are replaced after each drawing; (ii) when they are not replaced.

(i) Before the observed event there are n+1 hypotheses, equally likely; for the bag may contain 0, 1, 2, 3, n white balls. Hence following the notation of Art. 471,

...

Now the chance that the next drawing will give a white ball = Σ = Qr ;

and the value of numerator and denominator may be found by Art. 405.

If n is indefinitely large, the chance is equal to the limit, when n is in

r=nr - - k

Qr

k

The chance that the next drawing will give a white ball Σ

r=0 N

(-1) r

which is independent of the number of balls in the bag at first.

Example . A person writes n letters and addresses n envelopes; if the letters are placed in the envelopes at random, what is the probability that every letter goes wrong?

Let

Ип

...

denote the number of ways in which all the letters go wrong, and let abcd represent that arrangement in which all the letters are in their own envelopes. Now if a in any other arrangement occupies the place of an assigned letter b, this letter must either occupy a's place or some other.

(i) Suppose b occupies a's place. Then the number of ways in which all the remaining n-2 letters can be displaced is un-2, and therefore the umbers of ways in which a may be displaced by interchange with some one f the other n-1 letters, and the rest be all displaced is (n − 1) un−2·

(ii) Suppose a occupies b's place, and b does not occupy a's. Then in arrangements satisfying the required conditions, since a is fixed in b's place, the letters b, c, d, ... must be all displaced, which can be done in un-1 ways; therefore the number of ways in which a occupies the place of another letter but not by interchange with that letter is (1) by

from which, by the method of Art. 444, we find u„ — nun-1 = ( − 1)” (u, — u1). Also u1=0, u=1; thus we finally obtain

Now the total number of ways in which the n things can be put in n places is n; therefore the required chance is

The problem here involved is of considerable interest, and in some of its many modifications has maintained a permanent place in works on the Theory of Probability. It was first discussed by Montmort, and it was generalised by De Moivre, Euler, and Laplace.

*484. The subject of Probability is so extensive that it is impossible here to give more than a sketch of the principal algebraical methods. An admirable collection of problems, illustrating every algebraical process, will be found in Whitworth's Choice and Chance; and the reader who is acquainted with the Integral Calculus may consult Professor Crofton's article Probability in the Encyclopædia Britannica. A complete account of the origin and development of the subject is given in Todhunter's History of the Theory of Probability from the time of Pascal to that of Laplace.

The practical applications of the theory of Probability to commercial transactions are beyond the scope of an elementary treatise; for these we may refer to the articles Annuities and Insurance in the Encyclopædia Britannica.

*EXAMPLES. XXXII. e.

1. What are the odds in favour of throwing at least 7 in a single throw with two dice?

2. In a purse there are 5 sovereigns and 4 shillings. If they are drawn out one by one, what is the chance that they come out sovereigns and shillings alternately, beginning with a sovereign?

3. If on an average 9 ships out of 10 return safe to port, what is the chance that out of 5 ships expected at least 3 will arrive?

4. In a lottery all the tickets are blanks but one; each person draws a ticket, and retains it: shew that each person has an equal chance of drawing the prize.

5. One bag contains 5 white and 3 red balls, and a second bag contains 4 white and 5 red balls. From one of them, chosen at random, two balls are drawn: find the chance that they are of different colours.

6. Five persons A, B, C, D, E throw a die in the order named until one of them throws an ace: find their relative chances of winning, supposing the throws to continue till an ace appears.

7. Three squares of a chess board being chosen at random, what is the chance that two are of one colour and one of another?

8. A person throws two dice, one the common cube, and the other a regular tetrahedron, the number on the lowest face being taken in the case of the tetrahedron; find the average value of the throw, and compare the chances of throwing 5, 6, 7.

9. A's skill is to B's as 1 : 3; to C's as 3 : 2; and to D's as 4 : 3: find the chance that A in three trials, one with each person, will succeed twice at least.

10. A certain stake is to be won by the first person who throws an ace with an octahedral die: if there are 4 persons what is the chance of the last?

11. Two players A, B of equal skill are playing a set of games; A wants 2 games to complete the set, and B wants 3 games: compare their chances of winning.

12. A purse contains 3 sovereigns and two shillings: a person draws one coin in each hand and looks at one of them, which proves to be a sovereign; shew that the other is equally likely to be a sovereign or a shilling.

13. A and B play for a prize; A is to throw a die first, and is to win if he throws 6. If he fails B is to throw, and to win if he throws 6 or 5. If he fails, A is to throw again and to win with 6 or 5 or 4, and so on: find the chance of each player.

14. Seven persons draw lots for the occupancy of the six seats in a first class railway compartment: find the chance (1) that two specified persons obtain opposite seats, (2) that they obtain adjacent seats on the same side.

15. A number consists of 7 digits whose sum is 59; prove that the chance of its being divisible by 11 is

4

21

16. Find the chance of throwing 12 in a single throw with 3 dice,