For any discrete random variable $$X,$$ with probability mass function $$P\left( {X = j} \right) = {p_j},$$
$${p_j}\,\, \ge 0,\,j \in \left\{ {0,..........,\,\,\,N} \right\},$$ and $$\,\,\sum\limits_{j = 0}^N {{p_j} = 1,\,\,} $$ define the polynomial function $${g_x}\left( z \right) = \sum\limits_{j = 0}^N {{p_j}{z^j}} .$$ For a certain discrete random variable $$Y$$, there exists a scalar $$\beta $$ $$ \in \left[ {0,1} \right]$$ such that $${g_y}\left( z \right) = {\left\{ {1 - \beta + \left. {\beta z} \right)} \right.^N}.$$ The expectation of $$Y$$ is

$$P$$ and $$Q$$ are considering to apply for a job. The probability that $$P$$ applies for the job is $${1 \over 4},$$ the probability that $$P$$ applies for the job given that $$Q$$ applies for the job is $${1 \over 2},$$ and the probability that $$Q$$ applies for the job given that $$P$$ applies for the job is $${1 \over 3}.$$ Then the probability that $$P$$ does not apply for the job given that $$Q$$ does not apply for the job is

If a random variable $$X$$ has a Poisson distribution with mean $$5,$$ then the expectation $$E\left[ {{{\left( {X + 2} \right)}^2}} \right]$$ equals _________.

Consider the following experiment.
Step1: Flip a fair coin twice.
Step2: If the outcomes are (TAILS, HEADS) then output $$Y$$ and stop.
Step3: If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output $$N$$ and stop.
Step4: If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is $$Y$$ is (up to two decimal places) _____________.