Probability

If \(\mathrm{A}\) and \(\mathrm{B}\) are two events such that \(\mathrm{P}(\mathrm{A})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{1}{3}\), then the value of $$\mathrm{P}\left(\mathrm{A}^{\p...

A fair die is tossed twice in succession. If \(\mathrm{X}\) denotes the number of fours in two tosses, then the probability distribution of \(\mathrm{X}\) is given by

Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Then mean of number of tens is

Three critics review a book. For the three critics the odds in favor of the book are \(2: 5, 3: 4\) and \(4: 3\) respectively. The probability that the majority is in favor of the book, is given by

A player tosses 2 fair coins. He wins ₹5 if 2 heads appear, ₹ 2 if one head appears and ₹ 1 if no head appears. Then the variance of his winning amount in ₹ is :

The p.d.f. of a discrete random variable is defined as
$$\mathrm{f}(x)=\left\{\begin{array}{l} \mathrm{k} x^2, 0 \leq x \leq 6 \\ 0, \text { otherwise } \end{array}\right.$$
Then the value of \(F(4)\) (c.d.f) is

For a binomial variate \(\mathrm{X}\) with \(\mathrm{n}=6\) if \(P(X=4)=\frac{135}{2^{12}}\), then its variance is

A fair die is tossed twice in succession. If \(\mathrm{X}\) denotes the number of sixes in two tosses, then the probability distribution of \(\mathrm{X}\) is given by

In a game, 3 coins are tossed. A person is paid ₹ 7 /-, if he gets all heads or all tails; and he is supposed to pay ₹ 3 /-, if he gets one head or two heads. The amount he can expect to win on an average per game is ₹

If the sum of mean and variance of a Binomial Distribution is \(\frac{15}{2}\) for 10 trials, then the variance is

The three ships namely A, B and C sail from India to Africa. If the odds in favour of the ships reaching safely are \(2: 5,3: 7\) and \(6: 11\) respectively, then probability of all of them arriving safely is

The incidence of occupational disease in an industry is such that the workmen have a \(10 \%\) chance of suffering from it. The probability that out of 5 workmen, 3 or more will contract the disease is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of kings is

If \(P(A \cup B)=0.7, P(A \cap B)=0.2\), then \(P\left(A^{\prime}\right)+P\left(B^{\prime}\right)\) is

The probability distribution of a discrete random variable X is

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If the probability distribution function of a random variable X is given as

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A die is thrown four times. The probability of getting perfect square in at least one throw is

If \(\mathrm{P}(\mathrm{A})=\frac{3}{10}, \mathrm{P}(\mathrm{B})=\frac{2}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{5}\), then \(\mathrm{P}(\mathrm{A} / \mathrm{B}) \times \mathrm{P}(\mathrm{B} / \mathrm{A})=\)

Two dice are thrown simultaneously. If X denotes the number of sixes, then the expectation of X is

The probability that at least one of the events \(E_1\) and \(E_2\) occurs is 0.6. If the simultaneous occurrence of \(\mathrm{E}_1\) and \(\mathrm{E}_2\) is $$0.2, \mathrm{P}\left(\mathrm{E}_1^{\prime}\right)+\mathrm{P}\left(\mathrm{E}_2^{...

The probability distribution of a random variable X is

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A fair coin is tossed for a fixed number of times. If probability of getting 7 heads is equal to probability of getting 9 heads, then probability of getting 2 heads is

The probability distribution of the number of doublets in four throws of
a pair of dice is given by

A random variable X has the following probability distribution

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Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime number. The probability that the second guest to arrive is given a room which is a prime ...

For the probability distribution given by following

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Let two cards are drawn at random from a pack of 52 playing cards. Let X be the number of aces obtained. Then the value of E(X) is

A fair coin is tossed 100 times. The probability of getting a head for even number of times is

If the function defined by \(f(x)=K(x-x^2)\) if \(0 < x < 1=0\), otherwise is the p.d.f. of a r.v.X, then the value of \(P\left(X<\frac{1}{2}\right)\) is

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on
one face is

A man is known to speck truth 3 out of 4 times. He throws a die and reports that it is 6. Then the probability that it is actually 6 is

A random variable X \(\sim\) B (n, p), if values of mean and variance of X are 18 and 12 respectively, then n =

A coin is tossed three times. If X denotes the absolute difference between the number of heads and the number of tails, then P (X = 1) =

A random variable X has following distribution

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For two events \(\mathrm{A}\) and \(\mathrm{B}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{6}, \mathrm{P}(\mathrm{A})=\frac{1}{6}, \mathrm{P}(\mathrm{B})=\frac{2}{3}\), then \(\mathrm{A}\) and \(\mathrm{B}\) are

A fair coin is tossed 4 times. If \(X\) is a random variable which indicates number of heads, then \(\mathrm{P}[\mathrm{X}<3]=\)

First bag contains 3 red and 5 black balls and second bag contains 6 red and 4 black balls. A ball is drawn from each bag. The probability that one ball is red and the other is black, is

The distribution function \(F(X)\) of discrete random variable \(X\) is given by

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If the mean and variance of a binomial distribution are 4 and 2 respectively, then probability of getting 2 heads is

If \(\mathrm{X}\) is a random variable with p.m.f. as follows.
$$\begin{aligned}
\mathrm{P}(\mathrm{X}=\mathrm{x}) & =\frac{5}{16}, \mathrm{x}=0,1 \\
& =\frac{\mathrm{kx}}{48}, \mathrm{x}=2, \quad \text { then } \mathrm{E}(\mathrm{x})= \\
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A coin is tossed and a die is thrown. The probability that the outcome will be head or a number greater than 4 or both, is

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs selected at random from the lot and sent to retain store, then the probability that the store will receive at most one defective bulb is

In a meeting \(60 \%\) of the members favour and \(40 \%\) oppose a certain proposal. A member is selected at random and we take \(\mathrm{X}=0\) if he opposed and \(\mathrm{X}=1\) if he is in favour, then \(\operatorname{Var} \mathrm{X}=\)

It is observed that \(25 \%\) of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the probability that at least 5 of them will be solved is

an urn contains 9 balls of which 3 are red, 4 are blue and 2 are green. Three balls are drawn at random from the urn. The probability that the three balls have difference colours is

If \(X \sim B(4, p)\) and \(P(X=0)=\frac{16}{81}\), then \(P(X=4)=\)

Rajesh has just bought a VCR from Maharashtra Electronics and the shop offers after sales service contract for Rs. 1000 for the next five years. Considering the experience of VCR users, the following distribution of maintenance expenses for...

A random variable X has the following probability distribution

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Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number, is

Two unbiased dice are thrown. Then the probability that neither a doublet nor a total of 10 will appear is