## Class 12 Probability 7 important formulas

## Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. It is denoted by P(E|F), which represents the probability of event E occurring given that event F has already occurred.

**Properties of conditional probability:**

**P(E|F)**=**P(E ∩ F) / P(F)****P(E ∩ F)**=**P(E|F) × P(F)**

**Example**: If a card is drawn from a standard deck of 52 cards, what is the probability of drawing a heart given that the card drawn is red?

## Multiplication Theorem of Probability

The multiplication theorem of probability is used to find the probability of the intersection of two events. It states that the probability of the intersection of two events A and B is equal to the product of the probability of event A and the conditional probability of event B given that event A has occurred.

**Properties of the multiplication theorem of probability:**

**P(A ∩ B) = P(A) × P(B|A)**- If A and B are
**independent**events, then**P(A ∩ B) = P(A) × P(B)** - Multiplication rule of probability for more than two events If E, F and G are three events of sample space, we have

**P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)**

**Example**: What is the probability of drawing two aces in a row from a standard deck of 52 cards?

## Independent Events

Independent events are events that do not affect each other's occurrence. The probability of the intersection of two independent events is equal to the product of their individual probabilities.

**Definition**: Two events E and F are **independent** if and only if **P(E ∩ F) = P(E) × P(F).**

**Properties of independent events:**

- If E and F are independent events, then
**P(E|F) = P(E)** - If E and F are independent events, then
**P(F|E) = P(F)**

**Example**: Rolling a fair six-sided die twice. What is the probability of getting a 4 on the first roll and a 6 on the second roll?

## Theorem of Total Probability

The theorem of total probability is used to find the probability of an event by considering all possible ways in which it can occur. It states that the probability of an event A is equal to the sum of the probabilities of A given different mutually exclusive and exhaustive events E_{1}, E_{2}, ..., E_{n}, multiplied by their respective probabilities.

**Formula of Theorem of total probability:**

**Example**: A bag contains 3 red balls and 4 green balls. Two balls are drawn without replacement. What is the probability of drawing two green balls?

## Partitions of a Sample Space

In probability theory, a partition of a sample space is a collection of mutually exclusive and exhaustive events. It helps in simplifying the calculation of probabilities by breaking down the sample space into smaller, more manageable parts.

**A set of events E1, E2, ..., En is said to represent a partition of the sample space S if they are: **

**Pairwise disjoint**:**E**for all i ≠ j. This means that no two events in the partition can occur simultaneously._{i}∩ E_{j}= φ**Exhaustive:**The union of all events in the partition covers the entire sample space, i.e.,**E**_{1}∪ E_{2}∪ ... ∪ E_{n}= S.**Nonzero probabilities:**Each event in the partition has a probability greater than zero, i.e.,**P(E**for all i = 1, 2, ..., n._{i}) > 0

**Example:**

- A classic example of a partition is any nonempty event E and its complement E'. They satisfy E ∩ E' = φ (empty set) and E ∪ E' = S, fulfilling the criteria of a partition.
- From a Venn diagram, if E and F are any two events associated with a sample space S, then the set {E ∩ F', E ∩ F, E' ∩ F, E' ∩ F'} is a partition of the sample space.
- It's important to note that the partition of a sample space is not unique; multiple partitions can exist for the same sample space.

**Example**: Tossing two fair coins. Let A be the event of getting at least one head and B be the event of getting at least one tail. The sample space can be partitioned into three events: A ∩ B, A ∩ B', and A' ∩ B, where ' denotes the complement of an event.

## Bayes’ Theorem

Bayes' theorem is used to update the probability of an event based on new evidence or information. It relates the conditional probability of an event given another event to the conditional probability of the other event given the first event.

**Bayes’ Theorem:**

If $E,E,...,E$ are n non-empty events that constitute a partition of the sample space S, i.e., E$,E,...,E$ are pairwise disjoint, and $E∪E∪...∪E=S$, and $A$ is any event of nonzero probability, then for any $i=1,2,3,...,n$:

**Example: A** medical test for a certain disease is 95% accurate. The disease occurs in 1% of the population. If a person tests positive, what is the probability that they actually have the disease?

## Random Variable and Probability Distribution

**Random Variable: **A random variable is a variable that takes on different values based on the outcome( x_{1} , x_{2} , x_{3} , ... , x_{n} ) of a random event. It can be discrete or continuous.

Let $X$ be a random variable associated with a sample space $S$. The probability distribution of $X$ is often denoted by **$P(X=x)$**, representing the probability that the random variable takes on the

**Properties:**

**Domain [ $(X=x) ]$:**The set of all possible values that the random variable can take.**Probability Function [ $P(X=x) ]$ :**The function that assigns probabilities to each possible value of the random variable.

**Probability Distribution:**

A probability distribution describes how the probabilities are spread over the values of a random variable.

**Formula:**

For a discrete random variable $X$, the probability distribution is expressed as $P(X=x)$, where $x$ represents each possible value of $X$.

**Representation in Tabular Form:**

$x$ | $x$ | $x$ | $x$ | $...xn$ |
---|---|---|---|---|

$P(X=x_{i})$ | $p$ | $p$ | $p$ | $...pn$ |

**Properties:**

**Sum of Probabilities:**The sum of all probabilities in the distribution is equal to 1.**Each Probability is Non-Negative:**$P(X=x)≥0$ for all $x$.

## Mean of a Random Variable

The mean of a random variable is a measure of its central tendency. It represents the average value of the random variable over multiple trials.

For a discrete random variable $X$ with probability distribution $P(X=x) = Pi$, the **mean ($μ$)** is given by:

**$E(x)= μ=∑xipi$**

Properties of the mean of a random variable:

- The mean of a discrete random variable is calculated as the sum of the products of each possible value and its corresponding probability.
- The mean of a continuous random variable is calculated as the integral of the product of each possible value and its corresponding probability density function.,

**Example:**

Find Mean of above Probability Distribution .

**Example**: Let X be the number of goals scored by a soccer player in a match. What is the mean number of goals scored by the player?

**Example**: Let X be the number of heads obtained when flipping a fair coin three times. What is the probability distribution of X?

In conclusion, these formula notes provide a comprehensive overview of various topics in probability for 12th class mathematics. Understanding these concepts and their associated properties and examples will help students solve probability problems with ease.

12 Class Notes : Vector