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If two dice are rolled 12 times and getting a total greater than 4 is considered as a success, then which of the following statements are correct?

(A) The probability of getting a total greater than 4 in a single throw of the pair of dice is 5/6.

(B) Mean = 10

(C) Variance = 3/5

(D) The probability of getting a total less than or equal to 4 in a single throw of the pair of dice is 1/6.

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If the random variable X has the following probability distribution:

then K is equal to

A can solve 90% problems and B can solve 70% problems of the book. A problem is selected at random from the book. The probability that the problem is solved, is equal to

If A and B are two independent events, then which of the following is/are true?

(A) $P(A \cap B) = 0$

(B) $P(A \cup B) = 1 - P(A')P(B')$

(C) $P(A \cup B) = P(A)P(B)$

(D) $P(A \cap B) = P(A)P(B)$

Choose the correct answer from the options given below:

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be heart then the probability of the missing card to be a heart is:

Let E and F be two events such that $P(E) = \frac{1}{3}$, $P(F) = \frac{1}{4}$ and $P(E \cap F) = \frac{1}{5}$. Then the value of $P(F|E)$ is equal to

There are 50 telephone lines in an exchange. The probability that any one of them will be busy is 0.1, then the probability that all the lines are busy?

If a random variable y follows Poisson's distribution such that $P(X = 2) = 9 P(X = 4) + 90 P(X = 6)$, then sum of the mean and variance of X is

The mean of the number of heads in a simultaneous toss of three coins is

A die is thrown 6 times. If getting an odd number is a success, then the probability of at least 5 successes is

Let X denotes the number of hours a person uses a mobile and the probability distribution of X is as

Then the value of K is

A letter is known to have come either from KOLKATA or TATANAGAR. On the envelope just two consecutive letters TA are visible. The probability that letter has come from TATANAGAR is

Bag A contains 2 unbiased and 3 biased coins whereas Bag B contains 3 unbiased and 2 biased coins. A bag is selected at random and 2 coins are taken out simultaneously. The probability, that both coins are unbiased is:

If A and B are independent events, then which of the following statements are TRUE?

(A) $P(A \cap B) = P(A).P(B)$

(B) $P(A \cap B) = P(A) - P(B)$

(C) $P(A \cup B) = P(A) + P(B) - P(A).P(B)$

(D) $P(A \cap B) = P(A). P(B|A)$

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Match List-I with List-II

Let A and B be any two events

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If X is a normal variate with mean 16 and standard deviation 4, then the value of standard normal variate Z corresponding to X = 17 is:

In 5 trials of binomial distribution, the probability of 3 successes is 4 times the probability of 2 successes. The probability of success in each trial is:

The standard deviation of the number of tails in three tosses of a coin is:

Three bad eggs are mixed with 7 good ones. If two eggs are drawn one by one without replacement, then the probability distribution of the number (X) of bad eggs drawn is:

The random variable $x$ has a probability distribution $p(x)$ of the form $P(x=r)=\begin{cases} rk, & \text{if } r \le 2, \\ (r-1)k, & \text{if } 2<r\le 4, \\ 0, & \text{otherwise,} \end{cases}$ where $r \in \mathbb{N}\cup\{0\}$ and $k\in\mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers. Then:

(A) $k=\frac{1}{9}$

(B) $P(2\le x\le 3)=\frac{1}{2}$

(C) $P(x=4)=\frac{1}{3}$

(D) $P(x>1)=\frac{7}{8}$

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In class XII, suppose 5% of boys and 0.25% of girls are physically fit for a game. A fit student is selected at random from this class having same number of boys and girls. If the probability that the selected student is a girl is $\frac{m}{n}$, gcd(m, n) = 1, then m + n is equal to

A random variable X has the following probability distribution

Then P (X < 3) is

A bag contains 4 red and 6 green balls. A ball is drawn at random. Its colour is noted and is returned to the bag. One additional ball of the colour drawn is put in the bag. Again a ball is then drawn from the bag. The probability of this ball to be of green colour is

A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball drawn at random from one of the two boxes, then the probability of getting a black ball is

If the random variable X is normally distributed with mean 16 and standard deviation 4, then the standard normal variable Z corresponding to X = 17 is

A box contains 10 balls, each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, then the probability that none is marked with the digit 0 is:

In a game, a man wins ₹ 8 for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown. The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3. If X denotes the amount which the man wins or loses, then which of the following are correct?

(A) All the possible values of X are 8, 5, 2 and -1.

(B) The probability distribution of X is:

(C) The mean value of X is 75/16.

(D) The variance of X is 6615/256.

Choose the correct answer from the options given below:

If a random variable X has the following probability distribution:

then,

Match List-I with List-II

Choose the correct answer from the options given below:

If A speaks truth in 75% cases and B speaks truth in 80% cases, then the probability that they contradict each other in a statement, is:

If A and B are two events such that $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, $P(B|A) = \frac{1}{4}$, then $P(A|B)$ is: