Probability: Basic Concepts, Formulas with Examples

What is Probability?

What is Probability is a commonly asked question among students. Probability is a branch of mathematics that tells us how likely an event is to happen. It tells us how sure or unsure we are about a result. The value of probability always lies between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain to happen.

Probability is not just used in maths but also in our daily lives. From predicting the weather, playing cards, rolling a dice, or choosing matching socks, all these depend on probability. Even in sports results and political voting strategies, probability plays an important role in deciding possible outcomes.

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Probability Formula

Probability formula is the ratio of the number of favorable outcomes to the total number of possible outcomes. We write the probability formula like:

Probability of an event = Number of favorable outcomes ÷ Total number of outcomes

For example, if we toss a coin, there are 2 possible outcomes: heads or tails. The probability of getting heads = 1 (favorable outcome) ÷ 2 (total outcomes) = 1/2.

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Probability Theory Terminology

To understand the concept of probability better, it is important for you to first learn some important terms used in probability theory, as explained below:

• Experiment: An action or test that gives a result is called an experiment. For example, tossing a coin is an experiment.

• Sample Space: The list of all possible results of an experiment is called the sample space. For example, for tossing a coin, the sample space is Head, Tail.

• Favorable Outcome: The result we are looking for is called the favorable outcome. For example, if we roll a dice and want a 5, then getting a 5 is the favorable outcome.

• Trial: Each time we perform an experiment, it is called a trial. For example, every time we toss the coin, we are doing one trial.

• Random Experiment: An experiment where we know all possible results, but we are not sure which one will come, is called a random experiment. For example, rolling a dice.

• Event: Any result of a random experiment is called an event. For example, getting a 6 when a dice is rolled.

• Equally Likely Events: When different results have the same chance of happening, they are called equally likely events. For example, in a fair coin toss, getting a head or a tail are equally likely.

• Exhaustive Events: When we write all the possible results of an experiment, it is called exhaustive events. For example, when rolling a dice, the exhaustive events are 1, 2, 3, 4, 5, 6.

• Mutually Exclusive Events: Events that cannot happen at the same time are called mutually exclusive events. For example, while tossing one coin, we can either get a head or a tail, but not both together.

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Types of Probability

Types of probability in maths help you understand chances in different ways. Let’s learn about the four main types of probability one by one:

• Classical Probability is also known as theoretical probability. It is used when all results are equally likely. For example, when we roll a fair dice, there are 6 sides, and the chance of getting any one number is 1 out of 6.

• Empirical Probability is also called experimental probability. In this, we find probability by actually doing the experiment many times and checking the results. For example, if we toss a coin 50 times and it shows heads 28 times, then the probability of getting a head is 28 out of 50.

• Subjective Probability depends on someone’s own belief or guess, not on exact math. For example, a cricket fan may say, “My team has a high chance of winning today,” based on feelings, not numbers.

• Axiomatic Probability follows a set of fixed rules made by mathematicians. It says probability always lies between 0 and 1. An impossible event has probability 0, and a sure event has probability 1.

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Probability Rules

To solve questions related to probability, using some rules is important. These rules make it easy to find the chances of events happening. Find out the main probability rules in maths here:

Addition Rule

If we want to know the chance of event A or event B happening, we add their probabilities. But if both can happen together, we subtract that overlap once.
For example, when picking a card, the chance of getting a red card or a king uses this rule.

Complementary Rule

The probability of something not happening is equal to 1 minus the probability of it happening. For example, if the chance of rain today is 0.3, then the chance of no rain is 1 – 0.3 = 0.7.

Conditional Rule

Sometimes we already know that one event has happened, and we want to find the chance of another event. This is called conditional probability. For example, if we know a student is in Class 10, we may ask: What is the probability that the student also likes maths?

Multiplication Rule

If two events must happen together, we multiply their probabilities. For example, if we toss two coins, the chance of getting two heads can be found using the multiplication rule.

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Probability Sampling

Probability sampling is a way of choosing a small group (sample) from a big group (population) so that everyone has the same chance of being selected. It is just like writing everyone’s name on slips, putting them in a box, and picking some at random. This makes the process fair. There are four main types of probability sampling:

• Simple Random Sampling: Everyone has the same chance of being picked, just like a lucky draw.

• Systematic Sampling: We first pick a random starting point and then choose every nth person from the list. For example, start with the 3rd student and then select every 5th student after that.

• Stratified Sampling: The population is divided into groups (like boys and girls), and we take some samples from each group.

• Cluster Sampling: The population is divided into clusters (like schools or areas). Then we pick some clusters and study everyone in them.

Probability sampling is useful because it reduces bias and gives everyone a fair chance of being selected. It makes the results more reliable, allows calculation of sampling error, and helps researchers apply findings to the whole population confidently.

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Probability Examples With Solutions

Learning probability and applying the probability formula to real questions makes the concept much easier to understand. Here are some probability examples with step-by-step solutions that will help you practice and learn better.

Example 1: What is the probability of getting a sum of 8 when two dice are thrown?

Solutions:

• Total possible outcomes when two dice are thrown = 36.

• Favorable outcomes for sum 8 = (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes.

• Probability = Number of favorable outcomes ÷ Total outcomes = 5 ÷ 36

Hence, the probability of getting a sum of 8 is 5/36.

Example 2: In a bag, there are 7 red balls and 5 blue balls. One ball is picked randomly. Find the probability of getting a red ball.

Solutions:

• Favorable outcomes (red balls) = 7

• Total balls = 7 + 5 = 12

• Probability = 7 ÷ 12

Hence, the probability of picking a red ball is 7/12.

Example 3: There are 6 cards numbered 1, 2, 3, 4, 5, 6. Find the probability of picking a prime number first and a composite number second, without replacement.

Solutions:

• Prime numbers = {2,3,5} → 3 favorable outcomes

• Composite numbers = {4,6} → 2 favorable outcomes

• First pick: P(prime) = 3 ÷ 6 = 1 ÷ 2

• Second pick: P(composite) = 2 ÷ 5

• Total probability = 1/2 × 2/5 = 1/5

Therefore, the probability of picking a prime first and a composite second is 1/5.

Example 4: Find the probability of drawing a red face card from a standard deck of 52 cards.

Solutions:

• Total face cards = 12 (3 red face cards in hearts and 3 in diamonds → total 6 red face cards)

• Total cards = 52

• Probability = 6 ÷ 52 = 3 ÷ 26

Therefore, the probability of drawing a red face card is 3/26.

Also Read: Zero Divided by a Number

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