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

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

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 E and F be two events such that P(E) = \(\frac { 1 }{ 3 } ,P(F)=\frac { 1 }{ 4 } ,\) find \(P(E\cup F)\)  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 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 \(1\over2\)).

If P(A) = \(\frac { 7 }{ 13 } \), P(B) = \(\frac { 9 }{ 13 } \)and P​\((A\cap B)\)= \(\frac { 4 }{13 } \), evaluate P(A/B).

A die is thrown three times. Events A and B are defined as below:
A: 4 on the third throw
B: 6 on the first and 5 on the second throw.
Find the probability of A,given that B has already occured.

If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by 1– P(A′) P(B′)

Evaluate P(A ∪ B), if 2P(A) = P(B) = \(\frac { 5 }{ 13 } \) and P(A|B) = \(\frac { 2 }{ 5 } \)

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that
(i) the youngest is a girl,
(ii) at least one is a girl?

If A and B are two events such that:P(A) = \(\frac { 1 }{ 4 } \), P(B) = \(\frac { 1 }{ 2 } \)and \(P(A\cap B)=\frac { 1 }{ 8 } \) , Find P(not A and B).

If A and B are two independent events such that:
\(P(\overset { \_ }{ A } \cap B)=\frac { 2 }{ 15 } \) and  \(P(A\cap \overset { \_ }{ B) } =\frac { 1 }{ 6 } \) then find P(A) and P(B)

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 the distribution.

There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up tails 25% of the times and the third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows head, what is the probability that it was from the two headed coin?

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack, three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

There are three coins. First is a biased that comes up tails 60% of the times, second is also a biased coin that comes up heads 75% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the first coin?

Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution.