Introduction to Probability

Probability is the mathematical measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

1. Basic Definitions

• Experiment: A process or action that leads to one of several possible outcomes (e.g., tossing a coin).
• Sample Space (S): The set of all possible outcomes of an experiment. For a coin, S = {Head, Tail}.
• Event (E): A subset of the sample space (e.g., getting a Head).
• Theoretical Probability: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$.

2. Experimental Probability

This is based on actual trials or observations. It is the ratio of the number of times an event occurs to the total number of trials conducted. For example, if a die is thrown 100 times and '4' appears 20 times, the experimental probability of '4' is 20/100 = 0.2.

3. Addition Rule of Probability

The addition rule is used when we want to find the probability of one event OR another event occurring.

• Mutually Exclusive Events: These are events that cannot happen at the same time (e.g., rolling a 2 and a 5 on a single die). Formula: $P(A \cup B) = P(A) + P(B)$.
• Non-Mutually Exclusive Events: Events that can happen together (e.g., drawing a Red card and a King from a deck). Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

4. Multiplication Rule of Probability

This is used when we want to find the probability of one event AND another event occurring.

• Independent Events: The outcome of one event does not affect the other (e.g., tossing a coin twice). Formula: $P(A \cap B) = P(A) \times P(B)$.
• Dependent Events: The outcome of the first event affects the outcome of the second (e.g., drawing two cards without replacement). Formula: $P(A \cap B) = P(A) \times P(B|A)$.

5. Common Sample Spaces

• One Coin: {H, T} (2 outcomes)
• Two Coins: {HH, HT, TH, TT} (4 outcomes)
• One Die: {1, 2, 3, 4, 5, 6} (6 outcomes)
• Two Dice: (6 x 6 = 36 outcomes)
• Playing Cards: 52 cards (4 suits: Spades, Hearts, Diamonds, Clubs; 13 cards each).