Probability

Probability

1. Fundamentals of Probability (Recap and Algebra of Events)

Basic Terminology (Quick Recap)

Before diving into complex theorems, we must be absolutely clear on the foundational terms.

1. Random Experiment: An experiment where all possible outcomes are known, but the exact outcome cannot be predicted in advance.
2. Sample Space ($S$): The complete set of all possible outcomes of a random experiment.
3. Event ($E$): Any subset of the sample space. It can consist of one or more outcomes. If an outcome belongs to the subset $E$, we say the event has occurred.
4. Simple Event: An event containing exactly one outcome.
5. Compound Event: An event containing more than one outcome.
6. Impossible Event: An event that has absolutely no chance of occurring.
7. Sure Event (or Certain Event): An event that is absolutely guaranteed to happen.
8. Favorable Event (or Favorable Outcome): A favorable event refers to the specific results of an experiment that satisfy the condition you are looking for.

Algebra of Events

In Class 12, we heavily rely on set theory to understand probability. Let $A$ and $B$ be two events associated with a sample space $S$.

• Event "A or B" (Union): Denoted by $A \cup B$. It means at least one of the events $A$ or $B$ occurs.
• Event "A and B" (Intersection): Denoted by $A \cap B$. It means both events $A$ and $B$ occur simultaneously.
• Event "Not A" (Complement): Denoted by $A'$ or $A^c$ or $\bar{A}$. It means event $A$ does not occur.
  $$A' = S - A$$
• Event "A but not B" (Difference): Denoted by $A - B$ or $A \cap B'$. It means $A$ occurs but $B$ does not.

Crucial Types of Events

Understanding the distinction between these types of events is critical for solving complex problems later in the chapter.

1. Mutually Exclusive Events (Disjoint Events): Events that cannot occur at the same time. There is no common outcome between them.
   Mathematical Condition: $$A \cap B = \emptyset$$
2. Exhaustive Events: A set of events that together cover the entire sample space. At least one of these events is guaranteed to occur whenever the experiment is performed.
   Mathematical Condition: $$E_1 \cup E_2 \cup \dots \cup E_n = S$$
3. Mutually Exclusive and Exhaustive Events: A set of events that are both disjoint and cover the whole sample space. (This concept forms the basis of the Theorem of Total Probability).

Axiomatic Approach to Probability

Let $S$ be a finite sample space and $E$ be an event. The probability function $P$ satisfies the following axioms:

• Positivity: For any event $E$, $P(E) \ge 0$.
• Certainty: The probability of the entire sample space is 1.
  $$P(S) = 1$$
• Additivity: If $A$ and $B$ are mutually exclusive events, then the probability of their union is the sum of their individual probabilities.
  $$P(A \cup B) = P(A) + P(B)$$

Essential Formulas for Quick Recall

1. Probability of an Event: If there are $n(S)$ elementary events associated with a random experiment and $n(E)$ of them are favourable to an event $E$, then:
   $$P(E) = \frac{n(E)}{n(S)}$$
2. If $P(A) = 1$, then $A$ is called certain event.
3. If $P(A) = 0$, then $A$ is called impossible event.
4. Probability of Complementary Event: $$P(A) + P(A') = 1$$
5. General Addition Theorem: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
6. For Mutually Exclusive Events: $$P(A \cup B) = P(A) + P(B)$$
7. Addition Theorem for 3 Events:
   $$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$$

• Probability of occurrence of A only: $$P(A \cap B') = P(A) - P(A \cap B)$$
• Probability of occurrence of B only: $$P(A' \cap B) = P(B) - P(A \cap B)$$
• Probability of occurrence of exactly one of A and B:
  $$P(A \cap B') + P(A' \cap B) = P(A) + P(B) - 2P(A \cap B) = P(A \cup B) - P(A \cap B)$$

2. Conditional Probability

Definition and Formula

The conditional probability of an event $A$, given that event $B$ has already occurred, is denoted by $P(A|B)$.

If $A$ and $B$ are any two events associated with the same sample space:

Similarly, the probability of event $B$ occurring given that $A$ has already occurred is:

Properties of Conditional Probability

1. Certainty in the new space: The probability of the sample space $S$, or the given event $F$ itself, under the condition $F$ is always 1.
   $$P(S|F) = P(F|F) = 1$$
2. Addition Theorem for Conditional Probability: If $A$ and $B$ are any two events of $S$, and $F$ is an event such that $P(F) \neq 0$:
   $$P((A \cup B)|F) = P(A|F) + P(B|F) - P((A \cap B)|F)$$
3. Complementary Conditional Probability: The probability of an event not happening, given $F$, follows the standard complement rule.
   $$P(E'|F) = 1 - P(E|F)$$

3. Multiplication Theorem on Probability

The Theorem

If $A$ and $B$ are two events associated with a sample space $S$, then the probability of the simultaneous occurrence of events $A$ and $B$ is the product of the probability of one event and the conditional probability of the other.

• For two events A and B:
  $$P(A \cap B) = P(A) \cdot P(B|A)$$ (where $P(A) \neq 0$)
  or
  $$P(A \cap B) = P(B) \cdot P(A|B)$$ (where $P(B) \neq 0$)
• For three events A, B, and C:
  $$P(A \cap B \cap C) = P(A) \cdot P(B|A) \cdot P(C|(A \cap B))$$

Multiplication Rule for Independent Events

Two events $A$ and $B$ are said to be independent if the occurrence (or non-occurrence) of one does not affect the probability of the occurrence of the other. In this case, $P(A|B) = P(A)$ and $P(B|A) = P(B)$.

• Mathematical Condition for Independence:
  $$P(A \cap B) = P(A) \cdot P(B)$$
• For n Independent Events:
  $$P(E_1 \cap E_2 \cap \dots \cap E_n) = P(E_1) \cdot P(E_2) \cdot \dots \cdot P(E_n)$$

Important Points to Remember

1. Difference between Mutually Exclusive and Independent Events: Mutually exclusive events cannot happen together (Intersection is zero), while Independent events can happen together, but they do not influence each other.
2. If $A$ and $B$ are independent, then:
   • $A'$ and $B$ are independent.
   • $A$ and $B'$ are independent.
   • $A'$ and $B'$ are independent.

4. Theorem of Total Probability

Partition of a Sample Space

A set of events $E_1, E_2, \dots, E_n$ is said to represent a partition of the sample space $S$ if they are mutually exclusive (disjoint), exhaustive, and have non-zero probabilities.

Statement of the Theorem

Let {$E_1, E_2, \dots, E_n$} be a partition of the sample space $S$. For any event $A$ associated with $S$, the total probability of $A$ is given by:

In summation notation:

5. Bayes' Theorem

The Formula

If $E_1, E_2, \dots, E_n$ are $n$ mutually exclusive and exhaustive events, and $A$ is an event of non-zero probability, then the probability of occurrence of $E_i$ given $A$ is:

Where:

• $P(E_i)$ is the Prior Probability (probability of the cause before the experiment).
• $P(A|E_i)$ is the Likelihood (probability of the event given the cause).
• $P(E_i|A)$ is the Posterior Probability (probability of the cause after the event is observed).

Key Difference in Application

• Total Probability: Used when you want to find the probability of the "result" (e.g., What is the probability that a randomly drawn bulb is defective?).
• Bayes' Theorem: Used when the result is known and you want to find the "source" (e.g., Given that the bulb is defective, what is the probability it was produced by Machine A?).