Consider Guwahati $$(G)$$ and Delhi $$(D)$$ whose temperatures can be classified as high $$(H),$$ medium $$(M)$$ and low $$(L).$$ Let $$P\left( {{H_G}} \right)$$ denote the probability that Guwahati has high temperature. Similarly, $$P\left( {{M_G}} \right)$$ and $$P\left( {{L_G}} \right)$$ denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use $$P\left( {{H_D}} \right),$$ $$P\left( {{M_D}} \right)$$ and $$P\left( {{L_D}} \right)$$ for Delhi.

The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature.

	HD	MD	LD
HG	0.40	0.48	0.12
MG	0.10	0.65	0.25
LG	0.01	0.50	0.49

Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature $$\left( {{H_G}} \right)$$ then the probability of Delhi also having a high temperature $$\left( {{H_D}} \right)$$ is $$0.40;$$ i.e., $$P\left( {{H_D}|{H_G}} \right) = 0.40.$$ Similarly, the next two entries are $$P\left( {{M_D}|{H_G}} \right) = 0.48$$ and $$P\left( {{L_D}|{H_G}} \right) = 0.12.$$ Similarly for the other rows.

If it is known that $$P\left( {{H_G}} \right) = 0.2,\,\,$$ $$P\left( {{M_G}} \right) = 0.5,\,\,$$ and $$P\left( {{L_G}} \right) = 0.3,\,\,$$ then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature 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 _________.

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