Probability for Class 9

Probability is a fundamental concept in mathematics that quantifies the chance of various outcomes in uncertain or random situations. It provides a formal framework for reasoning about uncertainty and making informed decisions in the presence of chance. So, probability is a superpower concept that helps us understand chance in a world full of possibilities. In this chapter, we will delve into the fundamental principles of probability and its importance.

Probability

Probability is a measure of how likely something is to happen. We figure it out by looking at the number of times what we want to happen could occur compared to all the possible things that could happen.

When we have a list of all the things that could happen (we call this the sample space "S") and we are interested in a particular event (let's call it "E"), we can express the probability of E happening as P(E).

Terms Related to Probability

Experiment: An experiment is a trial that gives us a certain result. Tossing a coin or throwing a dice are examples of experiments.

Random Experiment: A random experiment is a kind of experiment where we can't be sure what will happen. It is like a trial and each trial can lead to one or more results.
There are two possible results when we toss a coin: heads (H) or tails (T) which is a random experiment.

Outcomes: An outcome is a possible result of a random experiment.
If we toss a coin, the outcomes are either heads or tails.

Sample Space: The sample space is all the possible outcomes we could get from a random experiment.
If we toss a coin, the sample space gets heads (H) or tails (T).
If we roll a die, the sample space is the numbers 1, 2, 3, 4, 5 or 6.

Event: An event is a group of outcomes from a random experiment.
Getting a head when tossing a coin or getting a number 5 when throwing a dice are examples of events.

Complementary Event: A complementary event is about what doesn't happen in relation to the sample space.
If an event E is about a particular outcome, then the complementary event (E') is about everything else (other than the event).
If throwing a dice to get 5 is an event E, then rolling a dice not to get 5 is the complementary event E'.

Here are some important things to know about probability:

1. In any experiment, if you know the probability of an event happening and the probability of its complement, then the sum of these two probabilities together is always 1. Mathematically, it is expressed as:

P(E) + P( E′) = 1

This reflects the idea that the event E or its complement E' must happen and there are no other possibilities. It is a fundamental principle in probability theory.

2. If the probability of an event happening P(E) is given, then the probability of a complementary event P(E′) is found by:

P( E′) = 1 − P(E)

3. Probability always lies between 0 and 1. Mathematically, it is expressed as:

0 ≤ P(E) ≤ 1

4. If something has zero chance of happening, the probability is 0. It means it is impossible:

P(E) = 0

5. If we are absolutely sure that something will happen, the probability is 1. It means it is certain:

P(E) = 1

Events Related to Probability

Tossing One Coin: When you toss a fair coin, there are two things that can happen: it can land on heads (H) or tails (T).

Tossing Two Coins Simultaneously: Now, if you toss two coins at the same time, there are four possibilities. Let's explore the outcomes:

1. Both coins show heads: HH
2. The first coin shows heads and the second coin shows tails: HT
3. The first coin shows tails and the second coin shows heads: TH
4. Both coins show tails: TT

Tossing Three Coins Simultaneously: When you toss three coins at the same time, there are eight possibilities to consider. Let's explore the outcomes:

1. All three coins show heads: HHH
2. All three coins show tails: TTT
3. Two coins show heads and the third coin shows tails: HHT
4. Two coins show tails and the third coin shows heads: TTH
5. The first coin shows heads and the other two coins show tails: HTT
6. The first coin shows tails and the other two coins show heads: THH
7. The second coin shows heads and the other two show tails: THT
8. The second coin shows tails and the other two show heads: HTH

In summary, when tossing three coins simultaneously, there are eight possible outcomes, each representing a different combination of heads (H) and tails (T).

Rolling One Dice: When you throw a dice, there are six potential outcomes. Each outcome corresponds to one of the six numbers on the faces: 1, 2, 3, 4, 5 and 6.

Understanding "At Least" and "At Most":

At Least: If you roll a dice and are asked for the probability of rolling at least 4, it means you want to consider all outcomes that are 4 or higher (5, and 6).

→ Favourable Outcomes: 4, 5, 6

At Most: On the other hand, If you roll a dice and are asked for the probability of rolling at most 4, you consider all outcomes that are 4 or lower (1, 2, and 3).

→ Favourable Outcomes: 1, 2, 3, 4

In simpler terms, "at least" is like saying "this much or more," and "at most" is like saying "this much or less."