# Probability

The word chance, possible, probably or likely convey some sense of uncertainty about the occurrence of some events. The entire world is filled with uncertainty. You make decisions affected by uncertainty virtually every day.

**Probability is the measure of the likeliness that an event will occur.**

**Class 9 Topics**

**Class 10 Topics**

**Class 11 Topics**

**Class 12 Topics**

- Conditional probability
- Multiplication theorem on probability
- Independent events
- Total probability theorem
- Bayes' theorem
- Random variable and its probability distribution
- Mean and variance of random variable
- Bernoulli trials and Binomial distribution

### Experiment

An operation which can produce some well-defined outcome is called an experiment. An experiment is defined as a process for which its result is well defined.

**Deterministic Experiment**

In any experiment if the result of an experiment is unique or certain, then the experiment is said to be deterministic in nature.

**Probabilistic Experiment**

If the result of the experiment is not unique and can be one of the several possible outcomes then the experiment is said to be probabilistic in nature.

**Random Experiment**

An experiment in which all possible out comes are known and the exact output cannot be predicted in advance is called a random experiment. The outcomes are known as **events**. For example,

- Rolling an unbiased dice
- Tossing a fair coin
- Drawing a card from a pack of well-shuffled cards
- Picking up a ball of certain color from a bag containing balls of different colors

### Sample Space

When you perform an experiment, then the set S of all possible outcomes is called the sample space. For example,

- In tossing a coin S = {H,T}
- If two coins are tossed then S = {HH,HT,TH,TT}
- In rolling a dice, S = {1,2,3,4,5,6}

### Events

**Simple Event**

A simple event (or elementary event) is the most basic possible outcome of a random experiment and it cannot be decomposed further.

**Equally Likely Events**

Events are said to be equally likely when there is no reason to expect any one of them rather than any one of the others. For example, when a die is thrown, any number 1 or 2 or 3 or 4 or 5 or 6 may occur. In this trial, the six events are equally likely.

When a coin is tossed, the events {head} and {tail} are equally likely.

**Exhaustive Events**

A set of events is said to be exhaustive if no event outside this set occurs and at least one of these events must happen as a result of an experiment. For example, When a die is rolled, the set of events {1, 2, 3}, {2, 3, 5}, {5, 6} and {4, 5} are exhaustive events.

**Mutually Exclusive Events (Disjoint Events)**

Two or more events are said to be mutually exclusive if they have no outcomes in common. They cannot occur simultaneously. For example, when you roll a die the events {1, 2, 3} and {4, 5, 6} are mutually exclusive events.

**Tossing of Coin**

When you toss a coin, then either a Head (H) or a Tail (T) appears.

**Rolling of Dice**

A dice is a solid cube with 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When you throw a die, the outcome is the number that appears on its upper face.

**Drawing a Card**

A pack of cards has 52 cards. It has 13 cards of each suit, namely spades, clubs, hearts and diamonds. Cards of spades and clubs are black cards. Cards of hearts and diamonds are red cards.

There are four honors of each suit. These are Aces, Kings, Queens and Jacks. These are called Face cards.

### Probability of Event

Any subset of a sample space is called an Event.

**Probability of Occurrence of an Event**

If in a random experiment, there are n mutually exclusive and equally likely elementary events in which n elementary events are favorable to a particular event E, then the probability of the event E is defined as P(E)

p(E) = n(E)/n(S)

**Results on Probability**

- P(S) = 1
- 0 < P(E) < 1
- Probability of an event lies between 0 and 1.
- Maximum value of probability of an event is one.

If A and B are mutually exclusive events, P(A ∪ B) = P(A) + P(B)

### Addition Theorem on Probability

If E_{1} and E_{2} are two events in a sample space S, then P (E_{1} ∪ E_{2}) = P (E_{1}) + P (E_{2}) - P (E_{1} ∩ E_{2})

If E_{1} and E_{2} are mutually exclusive events (disjoint), then P (E_{1} ∪ E_{2}) = P (E_{1}) + P (E_{2})

### Independent and Dependent Events

Two or more events are said to be independent if the happening or non-happening of any one does not depend (or not affected) by the happening or non-happening of any other. Otherwise they are called dependent events.

For example, suppose a card is drawn from a pack of cards and replaced before a second card is drawn. The result of the second drawn is independent of the first drawn. If the first card drawn is not replaced, then the second drawn is dependent on the first drawn.

If E_{1} and E_{2} are independent events, then P(E_{1} ∩ E_{2}) = P(E_{1}) × P(E_{2})