Chapter 13 Probability

Notes

1. Introduction to Probability

Probability is the study of chance. It tells us how likely an event is to occur.

For example, suppose you toss a coin. You may get a head or a tail, but you cannot predict the outcome with certainty. Probability helps us measure the chance of getting a particular outcome.

2. Random Experiment

A random experiment is an experiment whose outcome cannot be predicted with certainty in advance.

Examples:

• Tossing a coin.
• Throwing a die.
• Drawing a card from a well-shuffled pack of cards.

3. Sample Space

The set of all possible outcomes of a random experiment is called its sample space. It is denoted by S.

Example 1: When a die is thrown once:

\[S=\{1,2,3,4,5,6\}\]

Example 2: When a coin is tossed twice:

\[S=\{HH,HT,TH,TT\}\]

Here, H represents Head and T represents Tail.

4. Event

An event is a subset of the sample space. It is generally denoted by A, B, or E.

Example: When a die is thrown:

\[S=\{1,2,3,4,5,6\}\]

Let A be the event of getting an odd number.

\[A=\{1,3,5\}\]

Let B be the event of getting a prime number.

\[B=\{2,3,5\}\]

Impossible Event: An event that cannot occur, represented by \(\varnothing\).

Sure Event: An event that always occurs, represented by S.

5. Probability of an Event

When all outcomes are equally likely, the probability of an event E is:

\[\boxed{P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}}\]

Or,

\[\boxed{P(E)=\frac{n(E)}{n(S)}}\]

Important Properties:

\[0\leq P(E)\leq1\]

\[P(\varnothing)=0\]

\[P(S)=1\]

\[\boxed{P(E)+P(\overline E)=1}\]

Here, \(\overline E\) represents the complement of event E.

6. Mutually Exclusive Events

Two events A and B are called mutually exclusive if they cannot occur at the same time.

\[\boxed{A\cap B=\varnothing}\]

Therefore,

\[\boxed{P(A\cap B)=0}\]

Example: Getting a head and getting a tail in a single toss of a coin are mutually exclusive events.

7. Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of occurrence of the other.