Consider the following propositional statements:

$${\rm P}1:\,\,\left( {\left( {A \wedge B} \right) \to C} \right) \equiv \left( {\left( {A \to C} \right) \wedge \left( {B \to C} \right)} \right)$$
$${\rm P}2:\,\,\left( {\left( {A \vee B} \right) \to C} \right) \equiv \left( {\left( {A \to C} \right) \vee \left( {B \to C} \right)} \right)$$ Which one of the following is true?

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?

A logical binary relation $$ \odot $$, is defined as follows:

Let ~ be the unary negation (NOT) operator, with higher precedence then $$ \odot $$. Which one of the following is equivalent to $$A \wedge B?$$

When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting the number of tosses till the first Head appears, including the toss where the Head appears. Assuming that successive tosses are indepandent, the expected value of N is