Class 9 Mathematic Probability All Formula & Concept Notes | Formula Sheet

Here’s a detailed overview of the probability concepts and formulas typically covered in a Class 9 mathematics curriculum:

1. Basic Probability Concepts

Probability measures the likelihood of an event occurring. It is defined as:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

• Experiment: An action or process with uncertain outcomes (e.g., rolling a die).
• Outcome: A possible result of an experiment (e.g., rolling a 4).
• Event: A subset of outcomes (e.g., rolling an even number).

Example: For rolling a fair six-sided die:

• Total number of possible outcomes = 6 (1, 2, 3, 4, 5, 6)
• Probability of rolling a 3 = $\frac{1}{6}$

2. Types of Events

• Simple Event: An event that consists of only one outcome (e.g., rolling a 2).
• Compound Event: An event that consists of more than one outcome (e.g., rolling an even number).

3. Classical Probability

Classical probability is used when all outcomes are equally likely.

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

4. Experimental Probability

Experimental probability is based on the actual results of an experiment.

$P(E) = \frac{\text{Number of times event E occurs}}{\text{Total number of trials}}$

5. Theoretical Probability

Theoretical probability is calculated based on the possible outcomes in a perfect scenario, without conducting experiments.

6. Complementary Events

The probability of the complement of an event $E$ (i.e., the event that $E$ does not occur) is given by:

$P(E') = 1 - P(E)$

7. Addition Rule of Probability

For any two events $A$ and $B$:

• If A and B are mutually exclusive (cannot happen at the same time):

  $P(A \cup B) = P(A) + P(B)$

• If A and B are not mutually exclusive:

  $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

8. Multiplication Rule of Probability

For any two events $A$ and $B$:

• If A and B are independent (one event does not affect the other):

  $P(A \cap B) = P(A) \times P(B)$

• If A and B are dependent (one event affects the other):

  $P(A \cap B) = P(A) \times P(B|A)$

  where $P(B|A)$ is the conditional probability of $B$ given $A$.

9. Conditional Probability

The probability of an event $B$ occurring given that $A$ has occurred is:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

10. Probability Distribution

For a discrete random variable, the probability distribution is a list of all possible outcomes and their probabilities. The sum of the probabilities of all possible outcomes is 1.

11. Expected Value

The expected value (mean) of a discrete random variable $X$ is:

$E(X) = \sum (x_i \cdot P(x_i))$

where $x_i$ are the possible values of $X$ and $P(x_i)$ are their probabilities.

12. Basic Problems and Examples

1. Single Die Roll: Find the probability of rolling an odd number.
2. Card Draw: Find the probability of drawing an Ace from a standard deck of 52 cards.
3. Coin Toss: Find the probability of getting at least one head in two tosses.

These concepts and formulas form the basis of probability theory as typically introduced in Class 9.