Suppose $${X_i}$$ for $$i=1,2,3$$ are independent and identically distributed random variables whose probability mass functions are $$\,\,\Pr \left[ {{X_i} = 0} \right] = \Pr \left[ {{X_i} = 1} \right] = 1/2\,\,$$ for $$i=1,2,3.$$ Define another random variable $$\,\,Y = {X_1}{X_2} \oplus {X_3},\,\,$$ where $$ \oplus $$ denotes $$XOR.$$ Then $$\Pr \left[ {Y = 0\left| {{X_3} = 0} \right.} \right]$$ =________.

In the given matrix $$\left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 1 & 0 \cr 1 & 2 & 1 \cr } } \right],$$ one of the eigenvalues is $$1.$$ The eigen vectors corresponding to the eigen value $$1$$ are

The number of $$4$$ digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set $${1, 2, 3}$$ is____________.

Consider the following policies for preventing deadlock in a system with mutually exclusive resources.

$$\,\,\,\,\,\,\,{\rm I}.$$ $$\,\,\,\,\,\,$$ Processes should acquire all their resources at the beginning of execution. If
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ any resource is not available, all resources acquired so far are released
$$\,\,\,\,\,{\rm II}.$$ $$\,\,\,\,\,\,$$ The resources are numbered uniquely, and processes are allowed to request
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ for resources only in increasing resource numbers
$$\,\,\,{\rm III}.$$ $$\,\,\,\,\,\,$$ The resources are numbered uniquely, and processes are allowed to request
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ for resources only in decreasing resource numbers
$$\,\,\,{\rm IV}.$$ $$\,\,\,\,\,\,$$ The resources are numbered uniquely. A process is allowed to request only
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ for a resource with resource number larger than its currently held resources

Which of the above policies can be used for preventing deadlock?