ma1254 random processes :: unit 1 :: probability & random variable

U N IT III PRO BABILITY AND RANDO M VARIABLES PART A 1 . D efinesam ple space. 2 . D efinem utually exclusive events. 3 . D efine probability ofan event. 4 . State the axiom sofprobability. 5 . State addition law ofprobability. 6 . D efine conditionalprobability. 7 . State m ultiplication rule ofprobability. 8 . D istinguish betw een conditionaland unconditionalprobabilities.; 9 . State the theorem on totalprobability. 10. State Baye’stheorem . 11.IfA and B are m utually exclusive eventsw ith P(A )= 0.4 and P(B)= 0.5 , find P(A B). 12.LetA and B be independenteventsw ith P(A )= 0.5 and P(B )= 0.8 , find P( B A ). 13.IfA and B are m utually exclusive eventsw ith P(A )= 0.29and P(B)= 0.43, find P(A B ). 14.IfA and B are eventsw ith P(A )= 3/8 , P(B)= ½ and P(A B)= ¼ , find P(A c B c ). 15. IfA and B are eventsw ith P(A )= ¾ , P(B)= 5/8 prove thatP(A B) 3/8. 16. Prove thatthe probability ofan im possible eventiszero. 17. Prove thatP( A )= 1 – P(A ). 18.IF B A , prove thatP(B) P(A ). 19.IfA and B are independentevents, prove that A and B are also independentevents. 20.IfA and B are independentevents, prove thatA and B are also independentevents. 21.IfA and B are independentevents, prove that A and B are also independentevents. 22. From 21 ticketsm arked w ith 20 to 40 num erals, one isdraw n atrandom . Find the chance thatitisa m ultiple of5. 23. Ifyou flip a balanced coin,w hatistheprobability ofgetting atleastone head 24. A isknow n to hitthe targetin tw o outof5 shotsw hereasB isknow n to hitthe target in 3 outof4 shots. Find the probability ofthe targetbeing hitw hen they both try. 25. Fourpersonsare chosen atrandom from a group containing 3 m en,2 w om en and 4 children. Find the chancethatexactly tw o ofthem willbe children 26. The oddsin favourofA solving a m athem aticalproblem are 3 to 4 and the odds againstB solving the problem are 5 to 7.Find the probability thatthe w illbe solved by atleastone ofthem . 27. A die isloaded in such a w ay thateach odd num beristw ice aslikely to occuraseach even num ber.Find P(G ),w here G isthe eventthata num bergreaterthan 3 occurson a single rollofthe die. 28. Tow dice are throw n together. Find the probability thattotalofthe num berson the Random processes

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MA1254 Random Processes :: Unit 1 :: Probability & Random Variable

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RAJALAKSHMI ENGINEERING COLLEGE, THANDALAM II BIOMEDICAL ENGINEERING BM1201 RANDOM PROCESSES

UNIT III

PROBABILITY AND RANDOM VARIABLES PART A

1 . Define sample space. 2 . Define mutually exclusive events. 3 . Define probability of an event. 4 . State the axioms of probability. 5 . State addition law of probability. 6 . Define conditional probability. 7 . State multiplication rule of probability. 8 . Distinguish between conditional and unconditional probabilities.; 9 . State the theorem on total probability. 10. State Baye’s theorem. 11. If A and B are mutually exclusive events with P(A) = 0.4 and P(B)= 0.5 , find P(AB). 12. Let A and B be independent events with P(A) = 0.5 and P(B) = 0.8 , find P( BA ). 13. If A and B are mutually exclusive events with P(A) = 0.29and P(B)= 0.43, find

P(A B ). 14. If A and B are events with P(A) = 3/8 , P(B) = ½ and P(AB)= ¼ , find P(AcBc). 15. If A and B are events with P(A) = ¾ , P(B) = 5/8 prove that P(AB) 3/8. 16. Prove that the probability of an impossible event is zero.

17. Prove that P( A ) = 1 – P(A). 18. IF B A , prove that P(B) P(A). 19. If A and B are independent events , prove that A and B are also independent events.

20. If A and B are independent events , prove that A and B are also independent events. 21. If A and B are independent events , prove that A and B are also independent events. 22. From 21 tickets marked with 20 to 40 numerals, one is drawn at random. Find the chance that it is a multiple of 5. 23. If you flip a balanced coin, what is the probability of getting at least one head 24. A is known to hit the target in two out of 5 shots whereas B is known to hit the target in 3 out of 4 shots. Find the probability of the target being hit when they both try. 25. Four persons are chosen at random from a group containing 3 men,2 women and 4 children. Find the chance that exactly two of them will be children 26. The odds in favour of A solving a mathematical problem are 3 to 4 and the odds against B solving the problem are 5 to 7.Find the probability that the will be solved by at least one of them. 27. A die is loaded in such a way that each odd number is twice as likely to occur as each even number. Find P(G) ,where G is the event that a number greater than 3 occurs on a single roll of the die. 28. Tow dice are thrown together. Find the probability that total of the numbers on the

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