A sender (S) transmits a signal, which can be one of the two kinds: H and L with probabilities 0.1 and 0.9 respectively, to a receiver (R).

In the graph below, the weight of edge (u, v) is the probability of receiving v when u is transmitted, where u, v ∈ {H, L}. For example, the probability that the received signal is L given the transmitted signal was H, is 0.7.

If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is ______

For n > 2, let a {0, 1}ⁿ be a non-zero vector. Suppose that x is chosen uniformly at random from {0, 1}ⁿ.
Then, the probability that $$\sum\limits_{i = 1}^n {{a_i}{x_i}} $$ is an odd number is _______.

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x² + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) _____.

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 __________.