Consider the first-order logic sentence
$$\varphi \equiv \,\,\,\,\,\,\,\exists s\exists t\exists u\forall v\forall w$$ $$\forall x\forall y\psi \left( {s,t,u,v,w,x,y} \right)$$
where $$\psi $$ $$(𝑠,𝑡, 𝑢, 𝑣, 𝑤, 𝑥, 𝑦)$$ is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose $$\varphi $$ has a model with a universe containing $$7$$ elements.

Which one of the following statements is necessarily true?

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 _______.

Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + 3$$ for all $$n = 0,1,2,....?$$

Two people, $$P$$ and $$Q,$$ decide to independently roll two identical dice, each with $$6$$ faces, numbered $$1$$ to $$6.$$ The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by $$P$$ and $$Q.$$ Assume that all $$6$$ numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to $$3$$ decimal places) that one of them wins on the third trial is _____.