398

he

*477. In the preceding article it has been suppo have no knowledge of the event except the statement and B; if we have information from other sources probability of the truth or falsity of the statement, tl taken into account in estimating the probability of the various hypotheses.

For instance, if A and B agree in stating a fact, of which the a priori probability is P, then we should estimate the probability of the truth and falsity of the statement by

Ppp' and (1-P) (1 − p) (1 − p′) respectively.

Example There is a raffle with 12 tickets and two prizes of £9 and £3. A, B, C, whose probabilities of speaking the truth are,, respectively, report the result to D, who holds one ticket. A and B assert that he has won the £9 prize, and C asserts that he has won the £3 prize; what is D's expectation?

Three cases are possible; D may have won £9, £3, or nothing, for A, B, C may all have spoken falsely.

Now with the notation of Art. 472, we have the a priori probabilities

*478. With respect to the results proved in Art. 476, it should be noticed that it was assumed that the statement can be made in two ways only, so that if all the witnesses tell falsehoods they agree in telling the same falsehood.

If this is not the case, let us suppose that c is the chance that the two witnesses A and B will agree in telling the same falsehood; then the probability that the statement is true is to the probability that it is false as pp' to c (1-p) (1 − p').

As a general rule, it is extremely improbable that two independent witnesses will tell the same falsehood, so that c is usually very small; also it is obvious that the quantity c becomes smaller as the number of witnesses becomes greater. These considerations increase the probability that a statement asserted by two or more independent witnesses is true, even though the credibility of each witness is small.

Example. A speaks truth 3 times out of 4, and B 7 times out of 10; they both assert that a white ball has been drawn from a bag containing 6 balls all of different colours: find the probability of the truth of the assertion.

There are two hypotheses; (i) their coincident testimony is true, (ii) it is false.

for in estimating p, we must take into account the chance that A and B will both select the white ball when it has not been drawn; this chance is

35

Now the probabilities of the two hypotheses are as Pipi to P2P2, and therefore as 35 to 1; thus the probability that the statement is true is 36

*479. The cases we have considered relate to the probability of the truth of concurrent testimony; the following is a case of traditionary testimony.

If A states that a certain event took place, having received an account of its occurrence or non-occurrence from B, what is the probability that the event did take place?

The event happened (1) if they both spoke the truth, (2) if they both spoke falsely; and the event did not happen if only one of them spoke the truth.

Let p, p' denote the probabilities that A and B speak the truth; then the probability that the event did take place is

pp' + (1 − p) (1 − p),

and the probability that it did not take place is

*480. The solution of the preceding article is that which has usually been given in text-books; but it is open to serious objections, for the assertion that the given event happened if both A and B spoke falsely is not correct except on the supposition that the statement can be made only in two ways. Moreover, although it is expressly stated that A receives his account from B, this cannot generally be taken for granted as it rests on A's testimony.

A full discussion of the different ways of interpreting the question, and of the different solutions to which they lead, will be found in the Educational Times Reprint, Vols. xxvII. and XXXII

*EXAMPLES. XXXII. d.

1. There are four balls in a bag, but it is not known of what colours they are; one ball is drawn and found to be white: find the chance that all the balls are white.

2. In a bag there are six balls of unknown colours; three balls are drawn and found to be black; find the chance that no black ball is left in the bag.

3. A letter is known to have come either from London or Clifton; on the postmark only the two consecutive letters ON are legible; what is the chance that it came from London?

4. Before a race the chances of three runners, A, B, C, were estimated to be proportional to 5, 3, 2; but during the race 4 meets with an accident which reduces his chance to one-third. What are now the respective chances of B and C?

5. A purse contains n coins of unknown value; a coin drawn at random is found to be a sovereign; what is the chance that it is the only sovereign in the bag?

6. A man has 10 shillings and one of them is known to have two heads. He takes one at random and tosses it 5 times and it always falls head: what is the chance that it is the shilling with two heads?

7. A bag contains 5 balls of unknown colour; a ball is drawn and replaced twice, and in each case is found to be red: if two balls are now drawn simultaneously find the chance that both are red.

8. A purse contains five coins, each of which may be a shilling or a sixpence; two are drawn and found to be shillings: find the probable value of the remaining coins.

9. A die is thrown three times, and the sum of the three numbers thrown is 15: find the chance that the first throw was a four.

10. A speaks the truth 3 out of 4 times, and B 5 out of 6 times: what is the probability that they will contradict each other in stating the same fact?

11. A speaks the truth 2 out of 3 times, and B 4 times out of 5; they agree in the assertion that from a bag containing 6 balls of different colours a red ball has been drawn: find the probability that the statement is true.

12. One of a pack of 52 cards has been lost; from the remainder of the pack two cards are drawn and are found to be spades; find the chance that the missing card is a spade.

13. There is a raffle with 10 tickets and two prizes of value £5 and £1 respectively. A holds one ticket and is informed by B that he has won the £5 prize, while C asserts that he has won the £1 prize: what is A's expectation, if the credibility of B is denoted by, and that of C by ?

14. A purse contains four coins; two coins having been drawn are found to be sovereigns: find the chance (1) that all the coins are sovereigns, (2) that if the coins are replaced another drawing will give a sovereign.

15. P makes a bet with Q of £8 to £120 that three races will be won by the three horses A, B, C, against which the betting is 3 to 2, 4 to 1, and 2 to 1 respectively. The first race having been won by A, and it being known that the second race was won either by B, or by a horse D against which the betting was 2 to 1, find the value of P's expectation.

16. From a bag containing n balls, all either white or black, all numbers of each being equally likely, a ball is drawn which turns out to be white; this is replaced, and another ball is drawn, which also turns out to be white. If this ball is replaced, prove that the chance

1

of the next draw giving a black ball is (n − 1) (2n+1)−1,

17. If mn coins have been distributed into m purses, n into each, find (1) the chance that two specified coins will be found in the same purse; and (2) what the chance becomes when purses have been examined and found not to contain either of the specified coins.

18. A, B are two inaccurate arithmeticians whose chance of solving a given question correctly are and respectively; if they obtain the same result, and if it is 1000 to 1 against their making the same mistake, find the chance that the result is correct.

19. Ten witnesses, each of whom makes but one false statement in six, agree in asserting that a certain event took place; shew that the odds are five to one in favour of the truth of their statement, even Ithough the a priori probability of the event is as small as 59+1*

1

LOCAL PROBABILITY. GEOMETRICAL METHODS.

*481. The application of Geometry to questions of Probability requires, in general, the aid of the Integral Calculus; there are, however, many easy questions which can be solved by Elementary Geometry.

Example 1. From each of two equal lines of length l a portion is cut off at random, and removed: what is the chance that the sum of the remainders is less than l?

Place the lines parallel to one another, and suppose that after cutting, the right-hand portions are removed. Then the question is equivalent to asking what is the chance that the sum of the right-hand portions is greater than the sum of the left-hand portions. It is clear that the first sum is equally likely to be greater or less than the second; thus the required probability is

1

2

COR. Each of two lines is known to be of length not exceeding 1: the 1 chance that their sum is not greater than I is 2'

Example 2. If three lines are chosen at random, prove that they are just as likely as not to denote the sides of a possible triangle.

Of three lines one must be equal to or greater than each of the other two; denote its length by l. Then all we know of the other two lines is that the length of each lies between 0 and 1. But if each of two lines is known to be of random length between 0 and 7, it is an even chance that their sum is greater than l. [Ex. 1, Cor.]

Thus the required result follows.

Example 3. Three tangents are drawn at random to a given circle: shew that the odds are 3 to 1 against the circle being inscribed in the triangle formed by them.

P

Draw three random lines P, Q, R, in the same plane as the circle, and draw to the circle the six tangents parallel to these lines.

Н. Н. А.

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