GATE IT 2006

In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data indicates that if the temperature at noon is less than or equal to 25^{\circ}C, the probability that it will rain in the afternoon is 0.4. The temperature at noon is equally likely to be above 25^{\circ}C, or at/below 25^{\circ}C. What is the probability that it will rain in the afternoon on a day when the temperature at noon is above 25^{\circ}C?

For the set N of natural numbers and a binary operation f : N \times N \to N, an element z \in N is called an identity for f, if f (a, z) = a = f(z, a), for all a \in N. Which of the following binary operations have an identity?
i. f (x, y) = x + y - 3
ii. f (x, y) = \max(x, y)
iii. f (x, y) = x^y

In the automaton below, s is the start state and t is the only final state.

Consider the strings u = abbaba, v = bab, \text{ and } w = aabb. Which of the following statements is true?

In the context-free grammar below, S is the start symbol, a and b are terminals, and\epsilon denotes the empty string
S \rightarrow aSa \mid bSb \mid a \mid b \mid \epsilon
Which of the following strings is NOT generated by the grammar?

Which regular expression best describes the language accepted by the non-deterministic automaton below?