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)

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

Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.

A bag contain 2 red, 6 black and 8 green balls. A ball is drawn at random from the bag. Find the probabilty:
(a) a red ball
(b) a black ball
(c) a green ball
(d) a non-red ball

If E and F are two events such that \(P(E)=\frac { 1 }{ 4 } ,\) \(P(E)=\frac { 1 }{ 2 } \) and \( P(E\cap F)=\frac { 1 }{ 8 } \), find
(a) P(E or F)
(b) P(not E and not F).

If P(F) = 0.35 and P(E\(\cup\)F) = 0.85 and E and F are independent events. Find P(E).

If P(A)=\(\frac { 2 }{ 5 } ,P(B)=\frac { 1 }{ 3 } ,P(A\cap B)=\frac { 1 }{ 5 } ,\) then find \(P(\bar { A } /\bar { B } )\) .

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 \(1\over2\)).

Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?

A die is thrown. If E is the event ‘the number appearing is a multiple of 3’ and F be the event ‘the number appearing is even’ then find whether E and F are independent ?

Given that E and F are events such that:
P(E) = 0.6, P(F) = 0.3 and \(P(E\cap F)\) = 0.2 find P(E/F) and P(F/E).

If P(A) = \(\frac { 3 }{ 5 } \) and P(B) = \(\frac { 1 }{ 5 } \) find \(P(A\cap B)\) if A and B are independent events.

A die is tossed thrice. Find the probability of getting an odd number at least once.

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of success.

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 4 balls. Two balls are drawn at random from the urn (without replacement) and are found to be white, What is the probability that all the four balls in the urn are white?

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.