The value of the definite integral $$\int_1^e {\sqrt x \ln \left( x \right)dx} $$ is

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field $$F = xi + yj + zk$$ defined with respect to a Cartesian coordinate system having $$i, j$$ and $$k$$ as unit base vectors. $$$\int {\int\limits_S {{1 \over 4}\left( {F.n} \right)dA} } $$$

Where $$S$$ is the sphere, $$\,\,{x^2} + {y^2} + {z^2} = 1\,\,$$ and $$n$$ is the outward unit normal vector to the sphere. The value of the surface integral is

Let $$X$$ be a normal random variable with mean $$1$$ and variance $$4.$$ The probability $$P\left\{ {X < 0} \right\}$$ is

The probability that a student knows the correct answer to a multiple choice question is $${2 \over 3}$$. If the student does not know the answer, then the student guesses the answer. The probability of the guessed answer being correct is $${1 \over 4}$$. Given that the student has answered the question correctly, the conditional probability that the student knows the correct answer is