14 - PROBABILITY Page 1 ( Answers at the end of all questions )

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1 - PROBABILITY Page ( ) Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is [ AIEEE 00 ] ( ) A random variable X has Poisson distribution with mean. Then P ( x >. ) equals e 0 - e e [ AIEEE 00 ] ( ) Let A and B be two events such that P ( A B ) = P ( A B ) = and P ( A ) = where A stands for complement of event A. Then events A and B are equally likely and mutually exclusive equally likely but not independent independent but not equally likely mutually exclusive and independent [ AIEEE 00 ] ( ) Let x x.. x n be n observations such that possible value of n among the following is x i = 00 and x i = 0. Then a [ AIEEE 00 ] ( ) Probability that A speaks truth is while this probability for B is. The probability that they contradict each other when asked to speak on a fact is 0 0 [ AIEEE 00 ] ( ) The mean and variance of a random variable x having a binomial distribution are and respectively. Then P ( x = ) is [ AIEEE 00 ]

2 - PROBABILITY Page ( ) A random variable X has the following probability distribution. X : p ( X ) : For the events E = { X is a prime number } and F = { X < } the probability P ( E F ) is [ AIEEE 00 ] x + ( ) The events A B C are mutually exclusive events such that P ( A ) = - x - x P ( B ) = and P ( C ) =. The set of possible values of x are in the interval [ 0 ] [ AIEEE 00 ] ( ) Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is [ AIEEE 00 ] ( 0 ) The mean and variance of a random variable X having a binomial distribution are and respectively. Then P ( X = ) is [ AIEEE 00 ] ( ) The probabilities of a student getting Ist IInd and IIIrd division in an examination are respectively and. The probability that a student fails in the examination is [ AIEEE 00 ] ( ) A bag contains red and black balls. A second bag contains red and black balls. One bag is selected at random. If from the selected bag one ball is drawn then the probability that the ball drawn is red is [ AIEEE 00 ]

3 - PROBABILITY Page ( ) A box contains nails and 0 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random then the probability that it is rusted or a nail is [ AIEEE 00 ] ( ) A bag contains brown and white socks. A man pulls out two socks. The probability that both the socks are of the same colour is [ AIEEE 00 ] ( ) A -faced fair dice is rolled repeatedly till appears for the first time. The probability that the dice is rolled for even number of times is [ IIT 00 ] ( ) Three distinct numbers are chosen randomly from first 00 natural numbers then the probability that all are divisible by and both is [ IIT 00 ] ( ) Two numbers are chosen from { } one after another without replacement. Find the probability that the smaller of the two is less than. [ IIT 00 ] ( ) If P ( B ) = P ( A B C ) = and P ( A B C = then P ( B C ) is [ IIT 00 ] ( ) If the integers m and n are chosen at random between and 00 then the probability that the number of the form m + n is divisible by equals [ IIT ]

4 - PROBABILITY Page ( 0 ) The probabilities that a student passes in Mathematics Physics and Chemistry are m p and c respectively. Of these subjects the student has a % chance of passing in at least one a 0% chance of passing in at least two and 0% chance of passing in exactly two. Which of the following relations are true? p + m + c = 0 p + m + c = 0 pmc = 0 pms = [ IIT ] ( ) If from each of the three boxes containing white and black white and black white and black balls one ball is drawn at random then the probability that white and black ball will be drawn is [ IIT ] ( ) A fair coin is tossed repeatedly. If tail appears on first four tosses then the probability of head appearing on fifth toss equals [ IIT ] ( ) Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals [ IIT ] ( ) If E and F are events with P ( E ) P ( F ) and P ( E F ) > 0 then occurrence of E occurrence of F occurrence of F occurrence of E non-occurrence of E non-occurrence of F none of the above implications holds [ IIT ] ( ) There are four machines and it is known that exactly two of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is [ IIT ]

5 - PROBABILITY Page ( ) If E and F are the complementary events of the events E and F respectively and if 0 < P ( F ) < then P ( E / F ) + P ( E / F ) = P ( E / F ) + P ( E / F ) = P ( E / F ) + P ( E / F ) = P ( E / F ) + P ( E / F ) = [ IIT ] ( ) If for the three events A B and C P ( exactly one of the events A or B occurs ) = P ( exactly one of the events B or C occurs ) = P ( exactly one of the events C or A occurs ) = p and P ( all the three events occur simultaneously ) = p where 0 < p < then the probability of at least one of the three events A B and C occurring is p + p p + p p + p p + p [ IIT ] ( ) Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral equals 0 0 [ IIT ] ( ) The probability of India winning a test match against West Indies is /. Assuming independence from match to match the probability that in a match series India s second win occurs at the third test is [ IIT ] ( 0 ) If 0 < P ( A ) < 0 < P ( B ) < and P ( A B ) = P ( A ) + P ( B ) - P ( A ) P ( B ) then P ( B / A ) = P ( B ) - P ( A ) P ( A B ) = P ( A ) + P ( B ) P ( A B ) = P ( A ) P ( B ) P ( A / B ) = P ( A ) [ IIT ] ( ) An unbiased die with faces marked and is rolled four times. Out of four face values obtained the probability that the minimum face value is not less than and the maximum face value is not greater than is then 0 [ IIT ]

6 - PROBABILITY Page ( ) Let E and F be two independent events. If the probability that both E and F happen is and the probability that neither E nor F happens is then P ( E ) and P ( F ) respectively are [ IIT ] ( ) India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0 and are and 0.0 respectively. Assuming that the outcomes are independent the probability of India getting at least points is [ IIT ] ( ) For any two events A and B in a sample space A P ( A ) + P ( B ) - P B P ( B ) P ( B ) 0 is always true P ( A ) = P ( A ) - P ( A ) P (B ) does not hold P ( A B ) = - P ( A ) P (B ) if A and B are independent P ( A B ) = - P ( A ) P (B ) if A and B are disjoint [ IIT ] ( ) If E and F are independent events such that 0 < P ( E ) < and 0 < P ( F ) < then E and F are mutually exclusive E and F c ( the complement of event F ) are independent E c and F c are independent P ( E / F ) + P ( E c / F ) = [ IIT ] ( ) One hundred identical coins each with probability p of showing us heads are tossed once. If 0 < p < and the probability of heads showing on 0 coins is equal to heads showing on coins then the value of p is [ IIT ] ( ) For two events A and B P ( A B ) is not less than P ( A ) + P ( B ) - not greater than P ( A ) + P ( B ) equal to P ( A ) + P ( B ) - P ( A B ) equal to P ( A ) + P ( B ) + P ( A B ) [ IIT ]

7 - PROBABILITY Page ( ) The probability that at least one of the events A and B occur is 0.. If A and B occur simultaneously with probability 0. then P ( A ) + P ( B ) is ( e ) none of these [ IIT ] ( ) A student appears for tests I II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I II and III are p q and respectively. If the probability that the student is successful is then p = q = p = q = p = q = 0 p = q = ( e ) none of these [ IIT ] ( 0 ) Three identical dice are rolled. The probability that the same number will appear on each of them is [ IIT ] ( ) If M and N are two events the probability that exactly one of them occurs is P ( M ) + P ( N ) - P ( M N ) P ( M ) + P ( N ) - P ( M N ) P ( M c ) + P ( N c ) - P ( M c N c ) P ( M N c ) + P ( M c N ) [ IIT ] ( ) Fifteen coupons are numbered respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is is none of these [ IIT ] ( ) If A and B are two events such that P ( A ) > 0 and P ( B ) then P ( A / B ) is equal to - P ( A / B ) - P ( A / B ) - P ( A B ) P ( B ) P ( A ) P ( B ) [ IIT ]

8 - PROBABILITY Page ( ) Two fair dice are tossed. Let X be the event that the first die shows an even number and Y be the event that the second die shows an odd number. The two events X and Y are mutually exclusive independent and mutually exclusive dependent none of these [ IIT ] ( ) There are n persons ( n ) among whom are A and B who are made to stand in a row in random order. Probability that there are exactly r ( r n - ) persons between A and B is n - r n ( n - ) n - r - n ( n - ) ( n - r - ) n ( n - ) r n ( ) There are players from which four teams each of two players are formed. What is the probability that two specific players are in one team? ( ) A natural number is selected from the first 0 natural numbers. The probability that x - x + 0 < x - 0 is Answers 0 0 b c c b c d b a d a b d c d d d a a a b 0 0 a a b d b ad a c b cd a ad b ac bcd d abc c c b 0 0 acd c c d c d b

Intermediate Math Circles November 8, 2017 Probability II

Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B