EXERCISE

1. If A and B are two events such that P (A) = 0.5, P (B) = 0.6 and P (A \j B) = 0.8, find P (A/B) and P(B/A).

2. If A and B are two events such that P 04) = 0.4, P (B) = 0.6 and P (B/A) = 0.5, find P (A/B) and P(AkjB).

3. If P (not A) = 0.7, P (B) = 0.7 and P (B/A) = 0.5, find P (A/B) andP(Au B).

4. Given that P (A) = 0.4, P (B) = 0.7, P(A nB) = 0.2, find

(i) P(A/B), (ii) P(A'/B),  (iii), P(A/B') (iv) P(A'/B')

5. Given that P (A) = 0.8, P (A/B) = 0.8, P (A n B) = 0.5, find

(i) P(B)             (ii)  P(B/A) _m (iii) P(AuB) (iv) P(A/AuB)

(v) P(AnBfAuB) (v) P(AnB/B') (vii) P (An B/A)

6. Given that P (A) = 0.8, P (B) = 0.7, P(C) - 0.6, P (A/B) = 0.8, P (CIB) = 0.7, P(Ar\C) = 0.48, determine whether:

(i) A and B are independent, (ii) A and C are independent.

(iii) B and C are independent.

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7. Given that C and D are independent and that P (C/D) = y , P (C n D) = j, find (0 P(Q (ii) P(D)

8. The events A and B are such that P (A) = | P (B) = \ and P (A u B) = Show that A and B are

j o JU

neither mutually exclusive nor independent.

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9. The events A and B are such that P (A') = -, P (A/B) = j, P (A u B) = |, where A' denotes the event

"A does not occur". Find (0 P (A), (ii) P (An B), (iii) P (B), (/v) P (A/B'), where B' denotes the events "B does not occur". Determine whether A and B are independent.

10. A die is rolled. If the outcome is an odd number, what is the probability that it is prime.

11. A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once ?

12. Two dice are thrown. Find the probability that numbers appeared have a sum 8 if it is known, that the second die always exhibits 4.

13. A couple has 2 children. Find the probability that both are boys, if it is known that (i) one of hte children is a boy; (i'O the older child is a boy.

[Hint: S= B_lB₂,B_]G₂,G_lB₂,G_]G₂. Let A = both the children are boys, B = one of the children is a boy, C = the older child is a boy. Reqd. prob. = (i) P (A/B) (ii) P (AiQ.

14. A coin is tossed and if the coin shows head it is tossed again but if it shows a tail then a die is tossed. , , If 8 possible outcomes are equally likely, find the probability that the die shows a number greater than 4 if it is known that the first throw of the coin results in a tail.

< [Hint: 5= (//, H), (H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6).....

A = (T, 5UT, 6), B = (7, 1), (T, 2), (T, 3), (T, 4), (T, 5), (r, 6) "

P(A^B) Reqd. prob. = P (A/B) = ._pB) ] ■

15. In a class 40% students read Statistics, 25% Mathematics and 15% both Mathematics and Statistics. One student is selected at random. Find the probability:

(i) that he reads Statistics, if it is known that he reads Mathematics,' (n) that he reads Mathematics, if it is known that he reads Statistics. [Hint: n (A) = 40, n (B) = 25, n (A n B) = 15]

16. In a certain school, 20% students failed in English, 15% students failed in Mathematics and 10% students failed in both English and Mathematics. A student is selected at random. If he failed in English, what is the probability that he also failed in Mathematics ?

17. A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is back ?

18. An urn contains 5 white and 8 red balls. Two successive drawings of three balls at a time are made such that the balls are not replaced before the second draw. Find the probability that the first draw gives 3 white and second draw gives 3 red balls.

19. A bag contains 19 tickets, numbered from 1 to 19. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.

20. An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that at least one ball is black ?

[Hint. Required probability = 1 - Prob. (none is black)]

21. A bag contains 4 white, 7 black and 5 red balls. Three balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.