Chance and Probability

In daily life, we often experience situations where outcomes are uncertain. Some examples include:

Carrying a raincoat every day without rain, but forgetting it on a rainy day.
Preparing 4 out of 5 chapters for a test, only to find a major question from the unprepared chapter.
A usually punctual train arriving late on the day you reach early.

These situations indicate that some events have unequal chances of occurring.

However, in probability, we study experiments where all possible outcomes have equal chances of occurring. Examples include:

Tossing a coin, where the chances of getting heads or tails are equal.
Rolling a die, where each number (1 to 6) has an equal probability.
Drawing a card from a well-shuffled deck, where each card has the same likelihood of being picked.

Probability helps us measure the likelihood of an event using mathematical calculations.

Getting a Result (Random Experiment)

Before a cricket match, a coin is tossed to decide which team will bat first. The possible outcomes are Head or Tail.

If you toss the coin, can you control whether it lands on heads or tails? No, the result is purely random.

An experiment where the outcome cannot be predicted but has fixed possibilities is called a random experiment.

In this case, Head and Tail are the two possible outcomes.

Equally Likely Outcomes

When a coin is tossed multiple times, the number of heads and tails gradually become almost equal. Similarly, when a die is rolled many times, each number (1 to 6) appears nearly the same number of times.

This shows that in such experiments, all outcomes have the same chance of occurring. These are called equally likely outcomes.

For example:

Tossing a fair coin → Head and Tail are equally likely.
Rolling a fair die → Each number (1 to 6) is equally likely.

Linking Chances to Probability

When tossing a coin, the possible outcomes are Head or Tail, both equally likely.
- Probability of getting Head = 1/2
- Probability of getting Tail = 1/2
For a die with faces numbered 1 to 6, there are 6 equally likely outcomes.
- Probability of getting 2 = 1/6
- Probability of getting 5 = 1/6
- Probability of getting 7 = 0 (since 7 is not on the die)
- Probability of getting any number from 1 to 6 = 6/6 = 1 (certain event)

Probability helps measure how likely an event is to occur.

Outcomes as events

Each outcome of an experiment or a collection of outcomes make an event.

For example in the experiment of tossing a coin, getting a Head is an event and getting a
Tail is also an event.
In case of throwing a die, getting each of the outcomes 1, 2, 3, 4, 5 or 6 is an event.
Is getting an even number an event? Since an even number could be 2, 4 or 6, getting an even number is also an event.