College Algebra

Hence the probability of obtaining a white ball from the

In like manner, the probability of obtaining a white ball

561. Given the probability of the happening of an event in one trial, to find the probability of its happening exactly r times in n trials.

Let p be the probability of the happening of the event in one trial.

Then 1p is the probability of its failing (Art. 550). The probability that the event will happen in each of the first r trials, and fail in each of the remaining n ―r trials, is p' (1 - p)".

But the number of ways in which the event may happen exactly r times in n trials is equal to the number of combinations of n things taken r at a time, or

Hence the probability that the event will happen exactly r times in n trials is

n (n − 1) ... (n − r + 1) p′(1 − p)"→r.
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(1)

For example, putting r = 1, the probability that the event will happen exactly once in n trials is np(1 − p)"-1; putting r = 2, the probability that the event will happen exactly twice in n trials is "(n − 1) p2(1 − p)"-2; and so on.

In like manner, the probability that the event will fail exactly r times in n trials is

562. Given the probability of the happening of an event in one trial, to find the probability of its happening at least r times in n trials.

The event happens at least r times if it happens exactly n times, or fails exactly once, twice, n- - r times.

Therefore the probability that it happens at least r times is equal to the sum of the probabilities of its happening exactly n times, or failing exactly once, twice, ..., n-r times; which, by Art. 561, is

563. Example. A bag contains five tickets numbered 1, 2, 3, 4, 5. Five tickets are drawn at random, each being replaced before the next is drawn. Find the probability of drawing the ticket marked 1 exactly three times, and at least three times.

In this case, p = 1= = 3, n=5.

Then by Art. 561, (1), the probability of drawing the ticket marked 1 exactly three times is

And by Art. 562, the probability of drawing it at least three times is

EXAMPLES.

564. 1. If eight coins are tossed up, what is the chance that one and only one will turn up head?

2. Á purse contains one dollar and three dimes; another contains two dollars and four dimes; a third three dollars and one dime. What is the chance of obtaining a dollar by drawing a single coin from one of the purses taken at random ?

3. What is the probability of throwing exactly three aces in five throws with a single die?

4. What is the probability of throwing at least three aces in five throws with a single die?

5. If three cards are drawn from a pack, what is the chance that they will consist of a king, queen, and knave?

6. The probability that A can solve a certain problem is , and the probability that B can solve it is . Find the probability that the problem will be solved if both try.

7. If a coin is tossed up ten times, what is the chance that the head will present itself exactly five times?

8. A bag contains ten tickets numbered 0, 1, 2, ..., 9. If three tickets are drawn at random, what is the probability that their sum is 22?

9. Two bags contain each 4 black and 3 white balls. A ball is drawn at random from the first, and if it is white, it is put into the second bag, and a ball drawn at random from that bag. Find the chance of drawing two white balls.

10. If the odds are 5 to 3 against a person who is now 40 living till he is 65, and 11 to 6 against a person who is now 45 living till he is 70, what is the chance that at least one of these persons will be alive 25 years hence?

11. What is the probability of throwing aces with a pair of dice at least three times in four trials?

12. A bag contains 5 white and 8 black balls. Two drawings, each of 3 balls, are made, the balls first drawn not being replaced before the second trial; what is the chance that the first drawing will give 3 white, and the second 3 black balls?

13. A bag contains ten tickets, five numbered 1, 2, 3, 4, 5, and the rest blank. Three tickets are drawn at random, each being replaced before the next is drawn; what is the probability that their sum is 10?

14. A's skill at a game is two-thirds of B's. What is the chance that A wins at least two games out of five?

15. A bag contains 4 red, 3 white, and 2 black balls. ball is drawn and not replaced. Another ball is then drawn.

Find the chance that the two balls are of the same color.

16. A's skill at a game is double B's. What is the probability that A wins four games before B wins two?

17. A bag contains 6 red balls, 5 white balls, and 4 black balls. Four balls are drawn in succession, and are not replaced after being drawn. What is the chance that two of them are red, one white, and one black?

18. If one vessel out of every ten is wrecked, what is the chance that, out of five vessels expected, at least four will arrive safely?

19. A and B draw in succession, in the order named, from a purse containing three sovereigns and four shillings. Find their respective chances of first drawing a sovereign, the coins when drawn not being replaced.

20. A, B, and C draw in succession, in the order named, from a bag containing three white balls and five black balls. Find their respective chances of first drawing a white ball, the balls when drawn not being replaced.

XXXVIII. CONTINUED FRACTIONS.

565. A continued fraction is an expression of the form