Probability

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points \(0,\) \(1\) and \(2\) are \(0.45, 0.05\) and \(0.50\) respectively. Assuming that the outcomes are independent, the proba...

A lot contains \(50\) defective and \(50\) non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events \(A, B, C\) are defined as
\(A=\) (the first bulbs is defective)
\(B=\) (the second bulbs is non-def...

Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is ...............

For any two events \(A\) and \(B\) in a simple space

If the mean and the variance of binomial variate \(X\) are \(2\) and \(1\) respectively, then the probability that \(X\) takes a value greater than one is equal to ...............

In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is \(1/3\) and the probability that he copies the answer is \(1/6\). The probability th...

Let \(A\) and \(B\) be two events such that \(P\,\,\left( A \right)\,\, = \,\,0.3\) and \(P\left( {A \cup B} \right) = 0.8.\) If \(A\) and \(B\) are independent events then \(P(B)=\) ................

A is a set containing \(n\) elements. \(A\) subset \(P\) of \(A\) is chosen at random. The set \(A\) is reconstructed by replacing the elements of \(P.\) \(A\) subset \(Q\) of \(A\) is again chosen at random. Find the probability that \(P\)...

If the probability for \(A\) to fail in an examination is \(0.2\) and that for \(B\) is \(0.3\), then the probability that either \(A\) or \(B\) fails is \(0.5\)

If \(E\) and \(F\) are independent events such that \(0 < P\left( E \right) < 1\) and \(0 < P\left( F \right) < 1,\) then

A pair of fair dice is rolled together till a sum of either \(5\) or \(7\) is obtained. Then the probability that \(5\) comes before \(7\) is ...............

Suppose the probability for A to win a game against B is \(0.4.\) If \(A\) has an option of playing either a "best of \(3\) games" or a "best of \(5\) games" match against \(B\), which option should be choose so that the probability of his ...

For two given events \(A\) and \(B,\) \(P\left( {A \cap B} \right)\)

A box contains \(2\) fifty paise coins, \(5\) twenty five paise coins and a certain fixed number \(N\,\,\left( { \ge 2} \right)\) of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the tota...

One hundred identical coins, each with probability, \(p,\) of showing up heads are tossed once. If \(0 < p < 1\) and the probability of heads showing on \(50\) coins is equal to that of heads showing on \(51\) coins, then the value of \(p\)...

Urn \(A\) contains \(6\) red and \(4\) black balls and urn \(B\) contains \(4\) red and \(6\) black balls. One ball is drawn at random from urn \(A\) and placed in urn \(B\). The one ball is drawn at random from urn \(B\) and placed in urn ...

A man takes a step forward with probability \(0.4\) and backwards with probability \(0.6\) Find the probability that at the end of eleven steps he is one step away from the starting point.

If \({{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}\) and \(\,{{1 - 2p} \over 2}\) are the probabilities of three mutually exclusive events, then the set of all values of \(p\) is ..............

The probability that at least one of the events \(A\) and \(B\) occurs is \(0.6\). If \(A\) and \(B\) occur simultaneously with probability \(0.2,\) then \(P\left( {\overline A } \right) + P\left( {\overline B } \right)\) is

A lot contains \(20\) articles. The probability that the lot contains exactly \(2\) defective articles is \(0.4\) and the probability that the lot contains exactly \(3\) defective articles is \(0.6\). Articles are drawn from the lot at rand...

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 student passing in tests \(I\), \(II\) and \(III\) are \(p, q\)...

A box contains \(100\) tickets numbered \(1, 2, ....., 100.\) Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than \(10.\) The minimum number on them is \(5\) with probability .......

\(P\left( {A \cup B} \right) = P\left( {A \cap B} \right)\) if and only if the relation between \(P(A)\) and \(P(B)\) is .............

In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks the correct answers. The candidate decides to tick the answers at random, If he ...

Three identical dice are rolled. The probability that the same number will appear on each of them is

In a certain city only two newspapers \(A\) and \(B\) are published, it is known that \(25\)% of the city population reads \(A\) and \(20\)% reads \(B\) while \(8\)% reads both \(A\) and \(B\). It is also known that \(30\)% of those who rea...

If \(M\) and \(N\) are any two events, the probability that exactly one of them occurs is

A box contains \(24\) identical balls of which \(12\) are white and \(12\) are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the \(4\)th time on the \(7\)th ...

If the letters of the word "Assassin" are written down at random in a row, the probability that no two S's occur together is \(1/35\)

\(A, B, C\) are events such that
\(P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8\)
\(P\left( {AB} \right) = 0.08,P\left( {AC} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09\)
If $$P\left( {A \cup B \cup C} \right)...

Cards are drawn one by one at random from a well - shuffled full pack of \(52\) playing cards until \(2\) aces are obtained for the first time. If \(N\) is the number of cards required to be drawn, then show that $${P_r}\left\{ {N = n} \rig...

Fifteen coupons are numbered \(1, 2 ........15,\) 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 \(9,\) is

If \(A\) and \(B\) are two events such that \(P\left( A \right) > 0,\) and \(P\left( B \right) \ne 1,\) then \(P\left( {{{\overline A } \over {\overline B }}} \right)\) is equal to

\(A\) and \(B\) are two candidates seeking admission in \(IIT.\) The probability that \(A\) is selected is \(0.5\) and the probability that both \(A\) and \(B\) are selected is atmost \(0.3\). Is it possible that the probability of \(B\) g...

An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are \(0.4, 0.3, 0.2\) and \(0.1\) respectively. What is the pr...

For a biased die the probabilities for the different faces to turn up are given below :

This die tossed and you are told that either face \(1\) or face \(2\) has turned up. Then the probability that it is face \(1\) is ...............

The probability that an event \(A\) happens in one trial of an experiment is \(0.4.\) Three independent trials of the experiment are performed. The probability that the event \(A\) happens at least once is

Two events \(A\) and \(B\) have probabilities \(0.25\) and \(0.50\) respectively. The probability that both \(A\) and \(B\) occur simultaneously is \(0.14\). Then the probability that neither \(A\) nor \(B\) occurs is

Six boys and six girls sit in a row randomly. Find the probability that
(i) the six girls sit together
(ii) the boys and girls sit alternately.

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:

Balls are drawn one-by-one without replacement from a box containing \(2\) black, \(4\) white and \(3\) red balls till all the balls are drawn. Find the probability that the balls drawn are in the order \(2\) black, \(4\) white and \(3\) re...