Given P(A) = \(1\over2\), P(B) = \(1\over3\) and \(P(A\cap B)={1\over6}\)  Are the events A and B independent?

Bayes’ Theorem If E₁ , E₂ ,..., E_n are n non empty events which constitute a partition of sample space S, i.e. E₁ , E₂ ,..., E_n are pairwise disjoint and E₁∪ E₂∪ ... ∪ E_n = S and A is any event of nonzero probability, then
\(\mathrm{P}\left(\mathrm{E}_i \mid \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_i\right) \mathrm{P}\left(\mathrm{A}_{\mid} \mathrm{E}_i\right)}{\sum_{j=1}^n \mathrm{P}\left(\mathrm{E}_j\right) \mathrm{P}\left({\left.\mathrm{A} \mid E_j\right)}_1\right.} \text { for any } i=1,2,3, \ldots, n\)

Given P(A) = 0.2, P(B) = 0.3 and \(P(A\cap B)=0.3\) Find P(A/B)

Given P(A) = 0.4, P(B) = 0.7 and P(B/A) = 0.6, Find \(P(A\cup B)\)

Events E and F are given to be independent. Find P(F) if it is given that P(E) = 0.60 and P(E\(\cap\)F) = 0.35

One card is drawn is drawn from a pack of 52 cards. Find the probability of getting :
(a) a red card  
(b) a jack of hearts
(c) a black face card
(d) a king.

If P(E) = \(\frac { 6 }{ 11 } \), P(F) = \(\frac { 5 }{ 11 } \) and P(E \(\cup\)F) = \(\frac { 7 }{ 11 } \) then find (a) P(E/F), (b) P(F/E)

If P(F) =\(\frac { 1 }{ 2 } and\ P(F)=\frac { 1 }{ 5 } \) find \(P(\overline { E\cup F } )\) If E and F are independent events.

A box contains 50 bolts and 50 nuts. Half of the bolts and nuts are rusted. If two items are drawn with replacement, what is the probability that either both are rusted or both are bolts.

If each element of a second order determinant is either 0 or 1, what is the probability that the value of determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value assumed with probability \(1\over2\)).

A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared atlaest once?

Six balls are drawn successively from an urn containing 7 red and 9 black balls.Tell whether or not the trials of drawing black balls are Bernoulli trials when after each draws the ball drawn is:
(i) replaced
(ii) not replaced in the urn.

An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?

A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event '3 on the die’. Check whether A and B are independent events or not.

Find the probability distribution of number of doublets in three throws of apair of dice.

Out of 9 outstanding students of a school, there are 4 boys and 5 girls. A team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected.

12 cards, numbered 1 to 12, are placed in box mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also, find mean and variance of distribution.

Two numbers are selected at random (without replacement) from the first six positive integers.Let x denote the larger of the two numbers obtained.Find the probability distribution of random variable x and hence find the mean of the distribution.