Probability - Simple problems on single events

Random Experiment and Outcomes: A random experiment is a process where the result cannot be predicted with certainty. For example, tossing a fair coin is an experiment where the possible outcomes are Heads ($H$) or Tails ($T$). Visualize a coin spinning in the air; until it lands, the outcome is unknown, but we know it must be one of these two results.

Sample Space ($S$): The set of all possible outcomes of a random experiment is called the sample space. For a single throw of a six-sided die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$, which can be visualized as the six distinct faces of the cube labeled with dots.

Event ($E$): An event is a subset of the sample space consisting of one or more outcomes. For instance, in a die roll, the event 'getting an even number' corresponds to the set $E = \{2, 4, 6\}$. Visualize this as highlighting only the specific outcomes you are looking for within the entire sample space.

Equally Likely Outcomes: Outcomes are said to be equally likely if none of them is expected to occur in preference to others. In a well-shuffled deck of $52$ cards, every card has the same chance of being picked. Visualize a balanced scale where every individual outcome carries the same 'weight' or chance.

Probability Scale: The probability of an event is always a real number between $0$ and $1$ inclusive. Visualize a number line where $0$ represents an 'Impossible Event' (like rolling a $7$ on a standard die) and $1$ represents a 'Sure Event' (like rolling a number less than $7$ on a standard die).

Complementary Events: For every event $E$, there is an event 'not $E$' (denoted as $E'$ or $\bar{E}$) which occurs when $E$ does not occur. Visualize a Venn diagram where a rectangular box is the whole sample space and a circle inside represents $E$; the region outside the circle but inside the box is $E'$. The sum of their probabilities is always $1$.

Deck of Cards Composition: A standard deck contains $52$ cards divided into $4$ suits: Hearts, Diamonds, Spades, and Clubs. Visualize $13$ cards in each suit ($A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K$). Hearts and Diamonds are red ($26$ cards), while Spades and Clubs are black ($26$ cards). There are $12$ 'face cards' (Jacks, Queens, Kings) in total.

Theoretical Probability: This is the ratio of the number of outcomes favorable to an event to the total number of possible outcomes in a sample space. Visualize this as a fraction where the 'target' outcomes are the numerator and the 'total' possibilities are the denominator.