Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than $$100$$ hours given that it is of Type $$1$$ is $$0.7,$$ and given that it is of Type $$2$$ is $$0.4.$$ The probability that an LED bulb chosen uniformly at random lasts more than $$100$$ hours is _________.

Suppose that the eigen values of matrix $$A$$ are $$1, 2, 4.$$ The determinant of $${\left( {{A^{ - 1}}} \right)^T}$$ is _______.

Consider the following expressions:
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(i)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ false
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(ii)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$Q$$
$$\,\,\,\,\,\,\,\,\,\,\,$$ $$(iii)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ true
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(iv)$$ $$\,\,\,\,\,\,\,\,\,\,\,$$ $$P∨Q$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(v)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\neg QVP$$

The number of expressions given above that are logically implied by $$P \wedge \left( {P \Rightarrow Q} \right)$$) is _____________.

Let $$f(x)$$ be a polynomial and $$g\left( x \right) = f'\left( x \right)$$ be its derivative. If the degree of $$\left( {f\left( x \right) + f\left( { - x} \right)} \right)$$ is $$10,$$ then the degree of $$\left( {g\left( x \right) - g\left( { - x} \right)} \right)$$ is ___________.