PROBABILITY

1. A problem in mathematics is given to three students whose chances of solving it are 1/3, 1/5 and 1/6. What is the probability that atleast one of them solves the problem ?

2. ‘A, speaks truth in 60% cases and ‘B’ in 90% cases. In what percent of cases are they likely to contradict each other in stating the same fact.

3. A coin is tossed once. If it shows head, it is tossed again and if it shows tail , then a dice is tossed . let ‘E1’ be the event : ‘ the first throw of coin shows tail and ‘E2’be the event : the dice shows a number greater th an 4. Find P (E2/E1).

4. Two unbiased dice are thrown. Find the probability that the sum of numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3.

5. A pair of dice is thrown 3 times . if getting a total of 10 is considered a success, find the probability distribution of the number of successes. Also find mean and variance of the distribution.

6. A and B throw a die in turn till one of them gets 2 or 4 and wins the game. Find their respective probabilities of winning , if A starts the game ?

7. A bag contains 3 white and 5 black balls, and second bag contains 5 white and 3 black balls. One ball is transferred from first bag to second bag then a ball is drawn from the second bag find the probability that the ball drawn is white.

8. A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

9. A and B take turns in throwing two dice . tha first to throw a sum 10, being awarded. Show that if A throws first, their chances of winning are in the ratio 12 : 11.

10. There are two identical bags containing respectively 4 white and 3 red balls, 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. If the ball drawn is white , what is the probability that it is chosen from the first bag.

11. If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, find the distribution.12. If mean and variance of a binomial distribution are 4 and 4/3 respectively. Find P (X> 0).13. 10 % of the tools produced by a machine are defective. Find the probanility distribution of the

number of defective tools in a sample of 3 drawn at random.14. A can hit a target 4 out of 5 times. B can hit it 3 out of 4 times. C can hit it 2 out of 3 times. They

fire simultaneously. Find the probability that (i) any two out of A, B and C will hit the target. (ii) none of them will hit the target.

15. For three persons A,B and C the chances of being selected as a manager of a firm are in the ratio of 4 : 1 : 2 respectively. The respective probabilities for them to introduce a radical change in the marketing strategy are 0.3, 0.8 and 0.5. if the change does take place, find the probability that it is due to the appointment of B or C.

16. A box contains 6 red marbles numbered from 1 to 6 and 4 white marblesnumbered from 12 to 15. Find theprobability that the marble drawn at random is (i) white , (ii) white and odd numbered , (iii) even numbered, (iv) red or even numbered.

17. In a class 40% study statistics , 25% mathematics and 15% both mathematics and statistics. One student is selected at random. Find the probability (i) that he studies statistics, if it is given that he studies mathematics, (ii) that he studies mathematics, if it is known that he studies statistics.

18. By examining the chest X- ray, the probability that T.B. is detected when a person is actually suffering from it is 0.99. the probability that the doctor diagnosis incorrectly that a person has T.B. on the basis of X-ray is 0.001. in a certain city 1 in 1000 persons suffers from T.B.. a person is selected at random and is diagnosed to have T.B. what is the chance that he is actually has T.b.

19. A card from a deck of 52 cards is dropped. From the remaining cards, two cards are drawn and are found to be clubs. Find the probability that the dropped card is of jack.

20. A man takes a step forward with probability 0.4 and backward with probability 0.6. find the probability that at the end of 11 steps, he is just one step away from the starting point.

21. In a binomial distribution , the sum of mean and variance is 42 and their product is 360. Find the distribution.

22. Find the expectation of a b.d. B (4, 1/3).23. Two cards are drawn successively without replacement from a deck of 52 cardsfind the

probability distribution , expectation, and standard deviationof number of spades.24. Five dice are thrown 729 times. How many times do you expect to get 5 atleast once in dice. 25. Three cards are drawn successively from a well shuffled deck of 52 cards without

replacement . what is the probability that the first two cards are kings and third is an ace.

26. A problem in mathematics was given to solve to three students whose chances of solving it are ½, 1/3 and ¼ resp. what is the probability that the problem will be solved

27. A bag X contains 2 white and 3 red balls and bag Y contains 4 white and 5 red balls. What is the probability of drawing the red ball