Conditional Probability

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 at least one is a girl?

A black and a red dice are rolled. Find the conditional probability of obtaining the sum $\mathrm{}8$, given that the red die resulted in a number less than $\mathrm{}4$.

Two coins are tossed once. Find $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)$, If events $\mathrm{E}$ and $\mathrm{F}$ are defined as follows:
$\mathrm{}\mathrm{E}$: no tail appears, $\mathrm{}\mathrm{F}$: no head appears

A coin is tossed three times. Find $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)$. If events $\mathrm{E}$ and $\mathrm{F}$ are defined as follows:
$\mathrm{}\mathrm{E}$: at most two tails, $\mathrm{}\mathrm{F}$: at least one tail.

A coin is tossed three times. Find $\mathrm{P}\left(\mathrm{E}|\mathrm{F}\right)$, If events $\mathrm{E}$ and $\mathrm{F}$ are defined as follows:
$\mathrm{}\mathrm{E}$: at least two heads, $\mathrm{}\mathrm{F}$: at most two heads.

If $A$ and $B$ are any two events such that $P\left(A\right)+P\left(B\right)-P\left(A \mathrm{a}\mathrm{n}\mathrm{d} B\right)=P\left(A\right)$, then

If $\mathrm{}\mathrm{A}\mathrm{}$ and $\mathrm{}\mathrm{B}\mathrm{}$ are two events such that $\mathrm{}\mathrm{P}\left(\mathrm{A}\right)\ne 0\mathrm{}$ and $\mathrm{}\mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)=1$, then

An electronic assembly consists of two subsystems, say, $\mathrm{}\mathrm{A}\mathrm{}$ and $\mathrm{}\mathrm{B}$. From previous testing procedures, the following probabilities are assumed to be known:
$\mathrm{P}\left(\mathrm{A} \mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{s}\right)=0.2$
$\mathrm{P}\left(\mathrm{B} \mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{s} \mathrm{a}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{e}\right)=0.15$
$\mathrm{P}\left(\mathrm{A} \mathrm{and} \mathrm{B} \mathrm{fail}\right)=0.15$
Evaluate the following probability
$\mathrm{P}\left(\mathrm{A} \mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{s} \mathrm{a}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{e}\right)$

If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$)

Suppose we have four boxes. $\mathrm{}\mathrm{A},\mathrm{}\mathrm{B},\mathrm{}\mathrm{C}\mathrm{}$ and $\mathrm{}\mathrm{D}\mathrm{}$ containing colored marbles as given below:

How many times must a man toss a fair coin so that the probability of having at least one head is more than $\mathrm{}90\mathrm{\%}$?

If a leap year is selected at random, what is the chance that it will contain $\mathrm{}53\mathrm{}$ Tuesdays?

In a hurdle race, a player has to cross $\mathrm{}10\mathrm{}$ hurdles. The probability that he will clear each hurdle is $\frac{5}{6}$. What is the probability that he will knock down fewer than $\mathrm{}2\mathrm{}$ hurdles?

An urn contains $\mathrm{}25\mathrm{}$ balls of which $\mathrm{}10\mathrm{}$ balls bear a mark $\mathrm{X}$ and the remaining $\mathrm{}15\mathrm{}$ bear a mark $\mathrm{Y}$. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If $\mathrm{}6\mathrm{}$ balls are drawn in this way, find the probability that at least one ball will bear $\mathrm{Y}$ mark.

An urn contains $\mathrm{}25\mathrm{}$ balls of which $\mathrm{}10\mathrm{}$ balls bear a mark ‘ $\mathrm{X}$’ and the remaining $\mathrm{}15\mathrm{}$ bear a mark ‘ $\mathrm{Y}$’. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If $\mathrm{}6\mathrm{}$ balls are drawn in this way, find the probability that all will bear ‘ $\mathrm{X}$’ mark.

Suppose that $\mathrm{}5\mathrm{\%}\mathrm{}$ of men and $\mathrm{}0.25\mathrm{\%}\mathrm{}$ of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male?

A couple has two children, find the probability that both children are males, if it is known that at least one of the children is male.

$\mathrm{A}\mathrm{}$ and $\mathrm{}\mathrm{B}\mathrm{}$ are two events such that $\mathrm{}\mathrm{P}\left(\mathrm{A}\right)\ne 0$. Find $\mathrm{}\mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)$, if $\mathrm{}\mathrm{A}\mathrm{}$ is a subset of $\mathrm{}\mathrm{B}$

If $\mathrm{A}$ and $\mathrm{B}$ are events such that $\mathrm{P}\left(\mathrm{A}|\mathrm{B}\right)=\mathrm{P}\left(\mathrm{B}|\mathrm{A}\right)$, then

If $\mathrm{P}\left(\mathrm{A}\right)=\frac{1}{2}, \mathrm{P}\left(\mathrm{B}\right)=0$, then $\mathrm{}\mathrm{P}\left(\mathrm{A}|\mathrm{B}\right)$ is