Competitive exams test not just knowledge, but the ability to solve questions accurately and in the shortest possible time. To support this, EduRev has created a concise summary sheet of key formulae and shortcuts so that essential concepts can be revised quickly in one place. This document focuses on the important ideas in Probability. It provides clear shortcuts, useful formulae and quick revision points designed to strengthen practice and improve speed and accuracy while solving probability questions in competitive examinations.
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Basic definitions and notation
Random experiment: An action or process that leads to one of several possible outcomes, where the outcome cannot be predicted with certainty in advance.
Sample space (S): The set of all possible outcomes of a random experiment.
Event: Any subset of the sample space. An event occurs if the outcome of the experiment belongs to that subset.
Probability of an event A, P(A): A number between 0 and 1 (inclusive) that measures the likelihood that event A occurs. If all outcomes in the sample space are equally likely, P(A) = (number of favourable outcomes for A) / (total number of outcomes).
Key properties and rules
- Range: 0 ≤ P(A) ≤ 1 for any event A.
- Comparing events: If P(A) > P(B) then event A is more likely to occur than event B. If P(A) = P(B) then A and B are equally likely.
- Sure and impossible events: P(S) = 1 for the certain event S, and P(φ) = 0 for the impossible event φ.
- Complement rule: The probability that event E does not occur is P
- Addition rule (general): For any two events E and F, P(E ∪ F) = P(E) + P(F) - P(E ∩ F).
- Mutually exclusive events: If E and F are mutually exclusive (disjoint), then P(E ∩ F) = 0 and P(E ∪ F) = P(E) + P(F).
- Independent events: Two events E and F are independent if P(E ∩ F) = P(E)·P(F). For independent events the probability of all occurring is the product of their probabilities.
- Multiplication rule (independent events): To find the probability that two or more independent events occur in sequence, multiply the probability of each event occurring separately.
- Conditional probability: The probability of E given F (written P(E|F)) is P(E ∩ F) / P(F), provided P(F) > 0.
- Odds: If an event A has probability P(A), the odds in favour of A are P(A) : P(not A) = P(A) : [1 - P(A)]. When expressed in terms of counts, if an event has m favourable outcomes and n unfavourable outcomes, the odds in favour are m : n.
Worked examples and short solutions
Example: If a number is selected at random from the two-digit multiples of 6, what is the probability that it is divisible by 9?
Solution.
The two-digit multiples of 6 start at 12 and end at 96 with common difference 6.
Number of two-digit multiples of 6 = (96 - 12)/6 + 1 = 84/6 + 1 = 14 + 1 = 15.
Numbers that are multiples of both 6 and 9 are multiples of lcm(6,9) = 18. Two-digit multiples of 18 are 18, 36, 54, 72, 90, total 5.
Required probability = favourable / total
∴ Ans =
Example: If two dice are thrown, what is the probability of getting prime numbers on both dice?
Solution:
Prime faces on a single die are 2, 3 and 5 - three favourable outcomes out of six.
Probability prime on first die = 3/6 = 1/2.
Probability prime on second die = 3/6 = 1/2.
Dice throws are independent, so probability both prime =
Example: If three dice are thrown, what is the probability of getting a total (sum) equal to 4?
Solution:
All ordered triples (a, b, c) of die faces with a + b + c = 4 are: (1,1,2), (1,2,1), (2,1,1) - three outcomes.
Total ordered outcomes = 6·6·6 = 216.
Required probability =
Example: If two cards are drawn from a standard pack of 52 cards without replacement, what is the probability that the two cards are of different colours?
Solution:
First card can be any card. Given the first card, there are 26 cards of the opposite colour out of the remaining 51 cards.
Required probability = 
Example: There are 5 red, 4 green, 3 yellow and 8 white balls in a bag. If three balls are chosen at random without replacement, what is the probability that they are all of the same colour?
Solution:
Total number of ways to choose 3 balls from 20 = C(20,3) = 1140.
Number of ways to choose 3 red = C(5,3) = 10.
Number of ways to choose 3 green = C(4,3) = 4.
Number of ways to choose 3 yellow = C(3,3) = 1.
Number of ways to choose 3 white = C(8,3) = 56.
Total favourable = 10 + 4 + 1 + 56 = 71.
Example: In a game there are three rounds; the probabilities of winning the first, second and third rounds are
respectively. A prize will be given if a player wins all three rounds. What is the probability of winning the prize?Solution:
The player must win all three rounds.
For independent rounds, required probability = (probability of winning round 1) × (probability of winning round 2) × (probability of winning round 3).
∴ Required probability =
Question for Examples: ProbabilityTry yourself:Out of 13 applicants for a job, there are 5 women and 8 men. Two persons are to be selected for the job. The probability that at least one of the selected persons will be a woman is: View Solution
Short summary and tips
- Always identify the sample space and whether outcomes are equally likely before using counting formulae.
- Use complement probabilities when 'at least one' or 'none' cases are easier: P(at least one) = 1 - P(none).
- For sequential experiments, check whether events are independent or whether replacement affects probabilities.
- When counts are involved, use combinations/permutations for unordered/ordered selections respectively; for equally likely ordered outcomes, count favourable ordered outcomes and divide by total ordered outcomes.
- Learn common patterns (dice sums, card probabilities, urn problems) and practise reducing the sample space by symmetry or conditioning to save time.
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FAQs on Examples: Probability - CSAT Preparation - UPSC
| 1. What is probability theory? |  |
Ans. Probability theory is a branch of mathematics that deals with the study of uncertainty and randomness. It provides a framework for understanding and quantifying the likelihood of events occurring. This theory is used extensively in various fields, including statistics, economics, physics, and computer science.
| 2. What is the difference between theoretical probability and experimental probability? | |
Ans. Theoretical probability is based on mathematical calculations and is determined by analyzing the possible outcomes of an event. It is often expressed as a fraction or percentage. On the other hand, experimental probability is calculated based on actual observations or experiments. It involves conducting trials and recording the outcomes to estimate the probability of an event occurring.
| 3. How can probability be applied in real-life situations? | |
Ans. Probability theory has numerous practical applications in everyday life. It can be used to analyze weather forecasts, predict the likelihood of a stock market crash, determine the chances of winning a lottery, or assess the risk of certain diseases. Additionally, it helps in making informed decisions in fields such as insurance, gambling, and risk management.
| 4. What are the basic principles of probability theory? | |
Ans. The basic principles of probability theory include the Law of Large Numbers, which states that as the number of trials increases, the experimental probability approaches the theoretical probability. Additionally, the Addition and Multiplication Rules help determine the probability of compound events. The Complement Rule allows calculating the probability of an event not occurring, and the Conditional Probability Rule deals with events dependent on previous outcomes.
| 5. How can probability be used to analyze data? | |
Ans. Probability plays a crucial role in data analysis and statistical inference. By applying probability distributions, such as the normal distribution, researchers can assess the likelihood of certain values occurring and make inferences about a population based on sample data. Probability also helps in hypothesis testing, where the probability of obtaining certain results under the null hypothesis is evaluated to make conclusions about the population.
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