Probability

The conditional probability that \(X\ge6\) given \(X>3\) equals :

The probability that \(X\ge3\) equals :

The probability that X = 3 equals

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent is :

Consider the system of equations \(ax+by=0; cx+dy=0,\)
where \(a,b,c,d\) \(\in \left\{ {0,1} \right\}\)
STATEMENT - 1 : The probability that the system of equations has a unique solution is \({3 \over 8}.\) and
STATEMENT - 2 : The proba...

Let \({E^c}\) denote the complement of an event \(E.\) Let \(E, F, G\) be pairwise independent events with \(P\left( G \right) > 0\) and \(P\left( {E \cap F \cap G} \right) = 0.\) Then \(P\left( {{E^c} \cap {F^c}|G} \right)\) equals

Let H\(_1\), H\(_2\), ..., H\(_n\) be mutually exclusive and exhaustive events with P(H\(_i\)) > 0, i = 1, 2, ..., n. Let E be any other event with 0 < P(E) < 1.
Statement 1 : P(H\(_i\) | E) > P(E | H\(_i\)). P(H\(_i\)) for \(i=1,2,...,n\)....

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his w...

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his w...

Let \({H_1},{H_2},....,{H_n}\) be mutually exclusive and exhaustive events with \(P\left( {{H_1}} \right) > 0,i = 1,2,.....,n.\) Let \(E\) be any other event with \(0 < P\left( E \right) < 1.\)
STATEMENT-1:
$$P\left( {{H_1}|E} \right) > P\...

If \(n\) is even and E denotes the event of choosing even numbered urn \(\left(\mathrm{P}\left(u_{i}\right)=\frac{1}{n}\right)\),
then the value of \(\mathrm{P}(w / \mathrm{E})\) is :

If \(\mathrm{P}\left(u_{i}\right)=c\), where \(c\) is a constant then \(\mathrm{P}\left(u_{n} / w\right)\) is equal to:

If \(\mathrm{P}\left(u_{i}\right) \propto i\), where \(i=1,2,3, \ldots n\), then \(\lim_\limits{n \rightarrow \infty} \mathrm{P}(w)\) is equal to:

A six faced fair dice is thrown until \(1\) comes, then the probability that \(1\) comes in even no. of trials is

A person goes office either by car, scooter, bus or train, proability of which being \(\frac{1}{7}, \frac{3}{2}, \frac{2}{7}\) and \(\frac{1}{7}\), respectively. Probability that he reaches office late, if he takes car, scooter, bus or trai...

A person goes to office either by car, scooter, bus or train, the probability of which being \({1 \over 7},{3 \over 7},{2 \over 7}\) and \({1 \over 7}\) respectively. Probability that he reaches office late, if he takes car, scooter, bus o...

If three distinct numbers are chosen randomly from the first \(100\) natural numbers, then the probability that all three of them are divisible by both \(2\) and \(3\) is

\(A\) and \(B\) are two independent events. \(C\) is even in which exactly one of \(A\) or \(B\) occurs. Prove that \(P\left( C \right) \ge P\left( {A \cup B} \right)P\left( {\overline A \cap \overline B } \right)\)

A box contains \(12\) red and \(6\) white balls. Balls are drawn from the box one at a time without replacement. If in \(6\) draws there are at least \(4\) white balls, find the probability that exactly one white is drawn in the next two dr...

Two numbers are selected randomly from the set \(S = \left\{ {1,2,3,4,5,6} \right\}\) without replacement one by one. The probability that minimum of the two numbers is less than \(4\) is

If \(P\left( B \right) = {3 \over 4},P\left( {A \cap B \cap \overline C } \right) = {1 \over 3}\) and
\(P\left( {\overline A \cap B \cap \overline C } \right) = {1 \over 3},\,\,\) then \(P\left( {B \cap C} \right)\) is

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is \(p.\) If he fails in one of the exams then the probability of his passing in the next exam is \({p \over 2}\) otherwi...

\(A\) is targeting to \(B, B\) and \(C\) are targeting to \(A.\) Probability of hitting the target by \(A,B\) and \(C\) are \({2 \over 3},{1 \over 2}\) and \({1 \over 3}\) respectively. If \(A\) is hit then find the probability that \(B\) ...

A box contains \(N\) coins, \(m\) of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is \(1/2\), while it is \(2/3\) when a biased coin is tossed. A coin is drawn from the box at random a...

An unbiased die, with faces numbered \(1,2,3,4,5,6,\) is thrown \(n\) times and the list of \(n\) numbers showing up is noted. What is the probability that, among the numbers \(1,2,3,4,5,6,\) only three numbers appear in this list?

An urn contains \(m\) white and \(n\) black balls. A ball is drawn at random and is put back into the urn along with \(k\) additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probabil...

A coin has probability \(p\) of showing head when tossed. It is tossed \(n\) times. Let \({p_n}\) denote the probability that no two (or more) consecutive heads occur. Prove that \({p_1} = 1,{p_2} = 1 - {p^2}\) and $${p_n} = \left( {1 - p} ...

If the integers \(m\) and \(n\) are chosen at random from \(1\) to \(100\), then the probability that a number of the form \({7^m} + {7^n}\) is divisible by \(5\) equals

Eight players \({P_1},{P_2},.....{P_8}\) play a knock-out tournament. It is known that whenever the players \({P_i}\) and \({P_j}\) play, the player \({P_i}\) will win if \(i < j.\) Assuming that the players are paired at random in each rou...

The probabilities that a student passes in Mathematics, Physics and Chemistry are \(m, p\) and \(c,\) respectively. Of these subjects, the student has a \(75%\) chance of passing in at least one, a \(50\)% chance of passing in at least two,...

A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals

Let \({C_1}\) and \({C_2}\) be the graphs of the functions \(y = {x^2}\) and \(y = 2x,\) \(0 \le x \le 1\) respectively. Let \({C_3}\) be the graph of a function \(y=f(x),\) \(0 \le x \le 1,\) \(f(0)=0.\) For a point \(P\) on \({C_1},\) let...

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

If \(E\) and \(F\) are events with \(P\left( E \right) \le P\left( F \right)\) and \(P\left( {E \cap F} \right) > 0,\) then

Three players, \(A,B\) and \(C,\) toss a coin cyclically in that order (that is \(A, B, C, A, B, C, A, B,...\)) till a head shows. Let \(p\) be the probability that the coin shows a head. Let \(\alpha ,\,\,\,\beta\) and \(\gamma\) be, res...

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

If \(\overline E\) and \(\overline F\) are the complementary events of events \(E\) and \(F\) respectively and if \(0 < P\left( F \right) < 1,\) then

If from each of the three boxes containing \(3\) white and \(1\) black, \(2\) white and \(2\) black, \(1\) white and \(3\) black balls, one ball is drawn at random, then the probability that \(2\) white and \(1\) black ball will be drawn is

If \(p\) and \(q\) are chosen randomly from the set \(\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},\) with replacement, determine the probability that the roots of the equation \({x^2} + px + q = 0\) are real.

In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, \(3\) in the front and \(4\) at the back? How many seating arrangements are possible if \(3\) girls should sit together in a back row on adja...

For the three events \(A, B,\) and \(C,P\) (exactly one of the events \(A\) or \(B\) occurs) \(=P\) (exactly one of the two events \(B\) or \(C\) occurs)\(=P\) (exactly one of the events \(C\) or \(A\) occurs)\(=p\) and \(P\) (all the three...

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

The probability of India winning a test match against West Indies is \(1/2\). Assuming independence from match to match the probability that in a \(5\) match series India's second win occurs at third test is

Let \(0 < P\left( A \right) < 1,0 < P\left( B \right) < 1\) and
\(P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)P\left( B \right)\) then

Let \(A, B, C\) be three mutually independent events. Consider the two statements \({S_1}\) and \({S_2}\)
\({S_1}\,:\,A\) and \(B \cup C\) are independent
\({S_2}\,:\,A\) and \(B \cap C\) are independent
Then,

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards number...

If two events \(A\) and \(B\) are such that \(P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4\) and \(P\left( {A \cap {B^c}} \right) = 0.5,\) then \(P\left( {B/\left( {A \cup {B^c}} \right)} \right.\)$$\left. \, \righ...

An unbiased die with faces marked \(1,2,3,4,5\) and \(6\) is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than \(2\) and the maximum face value is not greater than \(5,\) is th...

Numbers are selected at random, one at a time, from the two- digit numbers \(00, 01, 02 ......, 99\) with replacement. An event \(E\) occurs if only if the product of the two digits of a selected number is \(18\). If four numbers are select...

\(E\) and \(F\) are two independent events. The probability that both \(E\) and \(F\) happen is \(1/12\) and the probability that neither \(E\) nor \(F\) happens is \(1/2.\) Then,