34. In how many ways can the letters of the word logarithms be arranged:

(i.) Without changing the order of the vowels? (ii.) Without changing the place of any vowels? (iii.) Without changing the relative order of vowels and consonants?

35. In how many ways can the letters of the word logarithms be arranged so that the second, fourth, and sixth places may be occupied by consonants?

36. In how many ways can 2 consonants and 1 vowel be chosen out of the letters of the word logarithms?

And in how many of these will the letter s occur? 37. In how many ways can a set of 12 black and 12 white pieces be placed on the black squares of a checkerboard?

38. Of the numbers from 10,000 to 100,000:

(i.) In how many is every digit an odd number? (ii.) In how many is every digit an even number? (iii.) In how many is there no digit lower than 6? (iv.) In how many is there no digit higher than 3? (v.) How many contain all the digits 1, 2, 3, 4, 5? (vi.) How many contain all the digits 0, 2, 4, 6, 8? 39. Out of 20 consecutive numbers in how many ways can two be selected whose sum shall be odd?

40. How many dominoes are there in a set numbered from double blank to double twelve?

41. At a post-office they keep 10 kinds of postage-stamps. In how many ways can a person :

(i.) Buy twelve stamps?

(ii.) Buy eight stamps?

(iii.) Buy eight different stamps?

CHAPTER XXIII.

CHANCE.

424. If an event can happen in a ways and fail in b ways, and if all these ways are equally likely to happen; if, also, only one can happen, and one must happen, then the mathematical probability or chance of the event happening is expressed by the fraction

α

a+b

I. The probability of an event happening is expressed by the fraction whose numerator is the number of favorable ways, and denominator the whole number of ways.

Thus, if 1 ball be drawn from a bag containing 3 white balls and 9 black balls, the chance of drawing a white ball is ; or, as it is expressed, one chance in four.

II. The probability of an event not happening is expressed by the fraction whose numerator is the number of unfavorable ways, and denominator the whole number of ways.

Thus, if a denote the number of favorable ways, and b the number. of unfavorable ways, then the fraction will express the proba

b

a+b

bility of the event not happening. If, for example, 1 ball be drawn from a bag containing 3 white and 9 black balls, the chance that it will not be a white ball is 2.

it is evident that the chance of an event happening, added to the chance of its not happening, is equal to 1; and, since

an event is certain to happen or not happen, it follows that in the theory of chances

III. Certainty is expressed by unity.

IV. The chance of an event not happening is found by subtracting from unity the chance that it does happen.

427. If the number of favorable ways is equal to the number of unfavorable ways, then

This is expressed by saying "the event is as likely to happen as not," or "there is an even chance for the event,” or "the odds are even" for and against the event.

Thus, in tossing a cent, there is an even chance that it will fall with the head up.

428. If ab, the chance of the event happening is >. This is expressed by saying "the event is probable," or "the odds are as a to b in favor of the event."

If ab, the chance of the event happening is <}. This is expressed by saying "the event is improbable,” or "the odds are as b to a against the event."

Thus, the odds are as 5 to 3 in favor of drawing a white ball at the first trial from a bag containing 5 white and 3 black balls.

Again, since a die has 6 faces, on one of which is an ace, the chance for an ace the first throw is ; and the odds are 5 to 1 against an ace.

429. It may be shown that,

V. If there are several events of which only one can happen, the chance that some one of them will happen is the sum of their respective chances of happening.

For, let a, b, c..... denote the number of ways favorable to the first, second, third..... event respectively; and let p denote the whole number of ways, all equally probable, and of which one, and only one, must happen. Then the chances of the first, second, third, events are

Since there are a + b + c +...... ways favorable to some one or other of the events happening, the chance in favor of some one or other of the events is

If, for example, a bag contain 3 white, 4 black, 5 red, and 6 green balls, the chance of drawing at the first trial a white or a black ball is 18+1=173; the chance of drawing a white or a black or a red ball is 13 + 1 + 1 = 1; the chance of drawing a white or a black or a red or a green ball is +18+18+18=1}=1; that is, certainty.

8

g

H

(1) When two dice are thrown, what is the chance of throwing double aces?

Each die may fall in any one of 6 ways; therefore both dice in 6×6=36 ways (2 403). Of these ways only one will give double aces. Hence, the chance of double aces

=

3. Ans.

(2) What is the chance of throwing doublets in a single throw with two dice?

The dice may fall in 36 ways. Of these, 6 will be doublets.
Hence, the chance of throwing doublets = = f. Ans.

(3) What is the chance of throwing a six and a five by a single throw of two dice?

The dice may fall in 36 ways. Of these ways the first die may turn up a six and the second a five, or the first may turn up a five and the second a six. Hence, the chance is 7. Ans.

=

(4) With two dice, what is the chance of making a throw so that one and only one die may turn up a five?

In 6 of the 36 possible ways one die will turn up a five, and the other also will turn up a five in 6 ways. One of these 12 ways will be double fives; so that there are 11 ways in which one die, and only one, will turn up a five, and the chance is . Ans.

(5) What is the chance of making a throw that will amount

to five?

Of the 36 possible ways, 1 and 4, 4 and 1, 2 and 3, 3 and 2 amount to five. Hence, the chance is = ‡. Ans.

(6) In a single throw with two dice, if the player may count the number on one of the dice, or the sum of the numbers on the two dice, what is the chance of throwing five?

The chance is }} + } = }}. Ans.

(7) If A's chance of winning a prize is, and B's 1, what is the chance that neither will obtain a prize?

The chance that one will win is += 4. Hence, the chance that neither will win is 1-4 17. Ans.

(8) If 4 cards are drawn from a pack of 52 cards, what is the chance that there will be one of each suit?

Four cards can be selected (8 415) from the pack in

But 4 cards can be selected so as to be one of each suit in 134 28,561 ways.

=

Hence, the chance is, nearly.

(8 404.)

(9) If 4 cards are drawn from a pack, what is the chance that they will be the 4 aces?

There are 4 = 24 ways of drawing the four aces, and 270,725 ways of drawing four cards. Hence, the chance is 27825, or 1 chance in 11,280.

(10) Three balls are to be drawn from an urn containing 5 black, 3 red, and 2 white balls. What is the chance of drawing 1 red and 2 black balls?

=

Three balls can be selected from the whole 10 in 10 × 9 × 8 1 x 2 x 3

120 ways. Also, 2 black balls can be selected from the 5 black 5 x 4 balls in 10 ways, and 1 red ball from the 3 red balls in 3