Basic Concepts of Probability (original) (raw)
Last Updated : 21 Apr, 2026
Probability is defined as the likelihood of the occurrence of any event. It gives a numerical value to the chance or likelihood of something happening. Probability is generally denoted by P(E), where E represents the event.
It is expressed as a number between 0 and 1:
- 0 means the event is impossible,
- 1 means the event is certain,
- Values between 0 and 1 represent partial chances.
**Concepts of Probability are used in various real life scenarios:
- **Stock Market : Investors and analysts use probability models to understand trends and patterns in the movement of stock prices.
- **Insurance: Insurance companies use probability models to estimate the likelihood of various events and to manage risks, which helps in setting premiums.
- **Weather Forecasting : Meteorologists use probability to predict the likelihood of weather events, such as rain, snow, storms, or temperature changes.
Formula for Probability
The probability formula is defined as the ratio of the number of favorable outcomes to the total number of outcomes.
**Probability of Event P(E) = [Number of Favorable Outcomes] / [Total Number of Outcomes]
The probability of an event E, denoted by P(E), is a number between 0 and 1 that represents the likelihood of E occurring.
- If P(E) = 0, the event E is impossible.
- If P(E) = 1, the event E is certain to occur.
- If 0 < P(E) < 1, the event E is possible but not guaranteed.
**Note: The sum of the probabilities of all events in a sample space is always equal to 1.
For example, When we toss a coin, there are only two possible outcomes: Heads (H) or Tails (T). However, if we toss two coins simultaneously, there will be four possible outcomes: (H, H), (H, T), (T, H), and (T, T).
Sample Space and Events Sample Space
- **Sample Space: The sample space, often denoted by S, is the set of all possible outcomes of an experiment. For example, when rolling a six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.
- **Event: An event is any subset of the sample space. It represents a specific outcome or a combination of outcomes. There are many different types of events in Probability such as Impossible and Sure Events, Mutually Exclusive Events, Exhaustive Events, Dependent and Independent Events. For example, rolling an even number E = {2, 4, 6} is an event in the context of rolling a die.
**Basic Probability Rules
- **Addition Rule: **P(A∪B) = P(A) + P(B) - P(A∩B), where A∪B denotes the union of events A and B.
- **Multiplication Rule for Independent Events: **P(A∩B) = P(A) × P(B), where A and B are independent events.
- **Complement Rule: **P(A ′) = 1 - P(A), where ′ A ′ denotes the complement of event A.
Probability Distribution
A probability distribution describes how the values of a random variable are spread or distributed. It tells us the probability of each possible outcome in a sample space and can be either discrete (for countable outcomes) or continuous (for measurable outcomes like height or time).
**Types of Probability Distributions:
- Bernoulli Distribution
- Binomial Distribution
- Poisson Distribution
- Uniform Distribution
- Normal Distribution (Gaussian)
- Exponential Distribution
Applications of Probability
Some of the common events for which we can use applications of probability to check the results are:
- Choosing a card from the deck of cards
- Flipping a coin
- Throwing a dice in the air
- Pulling a red ball out of a bucket of red and white balls
- Winning a lucky draw
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Solved Examples on Basic Concepts of Probability
**Example 1: There are 6 pillows in a bed, 3 are red, 2 are yellow , and 1 is blue. What is the probability of randomly picking a yellow pillow?
**Solution:
Probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e.
2/6 = 1/3
**Example 2: There is a container full of coloured bottles, red, blue, green, disposed of and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results:
- No. of blue bottles picked out: 300
- No. of red bottles: 200
- No. of green bottles: 450
- No. of orange bottles: 50
**a) What is the probability that Sumit will pick a green bottle?
For every 1000 bottles picked out, 450 are green.
Therefore,
P(green) = 450/1000 = 0.45
**b) If there are 100 bottles in the container, how many of them are likely to be green?
Experiment implies that 450 out of 1000 bottles are green.
Therefore,
Out of 100 bottles, 45 are green.
**Example 3: Find the probability of ‘getting 3 on rolling a die’.
**Solution:
Sample Space = S = {1, 2, 3, 4, 5, 6}
Total number of outcomes = n(S) = 6
Let A be the event of getting 3.
Number of favorable outcomes = n(A) = 1
i.e. A = {3}
Probability, P(A) = n(A)/n(S) = 1/6
Hence, P(getting 3 on rolling a die) = 1/6
**Example 4: A vessel contains 4 blue balls, 5 red balls, and 11 white balls. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?
**Solution:
Probability to get the first ball is red or the first event is 5/20
Since we have drawn a ball for the first event to occur, then the number of possibilities left for the second event to occur is 20 – 1 = 19
Hence, the probability of getting the second ball as blue or the second event is 4/19
Again with the first and second event occurring, the number of possibilities left for the third event to occur is 19 – 1 = 18
And the probability of the third ball is white or the third event is 11/18
Therefore, the probability is 5/20 x 4/19 x 11/18 = 220/6840 = 11/342 = 0.032
We can express it as: P = 3.2%.
Practice Problem Based on Probability
**Question 1. A jar contains 8 green marbles, 5 red marbles, and 7 blue marbles. If you randomly pick one marble, what is the probability of selecting a red marble?
**Question 2. If you roll two six-sided dice, what is the probability of rolling a sum of 8? (Use the appropriate probability rules and sample space to calculate your answer.)
**Question 3. In a survey of 1000 students, it was found that 600 prefer the GFG Online Gate batch, 300 prefer the Offline Gate batch, and 100 like both the Online and Offline Gate batches. What is the probability that a randomly selected student from this survey likes either the Online or Offline Gate batch?
**Question 4. A student has a 75% chance of passing an exam. What is the probability that the student will **not pass the exam?
**Answer :-
**1. 0.25
**2. 5/36
**3. 0.8
**4. 0.25