Probability - Simple Problems on Calculating Probability

Theoretical Probability: The probability of an event $E$ is defined as the ratio of the number of outcomes favorable to $E$ to the total number of equally likely outcomes in the sample space. Visually, imagine a circle representing all possible outcomes; the event $E$ is a shaded slice of that circle.

Sample Space and Outcomes: The sample space is the set of all possible results of a random experiment. For example, when tossing two coins, the visual representation is a tree diagram with four branches: $HH$, $HT$, $TH$, and $TT$. Each unique result is an 'outcome'.

Equally Likely Outcomes: Outcomes are equally likely if each has the same chance of occurring. For instance, a fair six-sided die, visually represented as a cube with dots $1$ to $6$, has a $\frac{1}{6}$ probability for each face.

Range of Probability: The probability of any event $E$ is a real number such that $0 \le P(E) \le 1$. On a horizontal number line (a visual probability scale), $0$ represents an 'Impossible Event' (far left), $0.5$ represents an 'Even Chance' (middle), and $1$ represents a 'Sure Event' (far right).

Complementary Events: For any event $E$, the event 'not $E$' (denoted as $\bar{E}$) is called its complement. Together, they cover the entire sample space. Visually, if $E$ is a circle inside a rectangular box (the sample space), then $\bar{E}$ is everything inside the box that is outside the circle.

Elementary Events: An event having only one outcome of the experiment is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is exactly $1$.

Playing Cards Composition: A standard deck contains $52$ cards divided into $4$ suits: Spades and Clubs (Black), and Hearts and Diamonds (Red). Each suit has $13$ cards. Visually, there are $12$ 'face cards' in a deck (Kings, Queens, and Jacks), which are often depicted with illustrations rather than numbers.