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) _____.

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 _____.