Probability Notes for Competitive Exams

Probability measures how likely an event is to happen. In daily life we use it without noticing: a 70% chance of rain, the chance of winning a toss, or the odds of a team winning a match. In competitive exams, probability questions are short, logical and scoring. They appear in bank PO and clerk exams, SSC CGL Tier 2, CDS and occasionally in UPSC CSAT, where they often look like simple reasoning puzzles. Probability builds directly on counting, so a basic grip on permutation and combination makes this chapter much easier.

Quick Facts: Probability at a Glance

• Basic formula: P(E) = Number of favourable outcomes ÷ Total number of outcomes.
• Range: 0 ≤ P(E) ≤ 1. P = 0 means impossible; P = 1 means certain.
• Complement rule: P(not E) = 1 − P(E).
• Addition rule: P(A or B) = P(A) + P(B) − P(A and B).
• Multiplication rule (independent events): P(A and B) = P(A) × P(B).

1. Basic Terms

2. Coins

A coin has two outcomes, so n coins (or one coin tossed n times) have 2ⁿ outcomes.

Example: Three coins are tossed. P(exactly 2 heads) = 3/8 (HHT, HTH, THH). P(at least one head) = 1 − P(no head) = 1 − 1/8 = 7/8.

Shortcut: P(exactly r heads in n tosses) = nCr ÷ 2ⁿ. For 2 heads in 4 tosses: 4C2 ÷ 16 = 6/16 = 3/8.

3. Dice

A die has 6 faces (1 to 6). Two dice give 6 × 6 = 36 outcomes; three dice give 216.

Example: P(sum of 8 with two dice) = 5/36. P(doublet, i.e. same number on both) = 6/36 = 1/6.

Example: P(sum is a prime number) with two dice: sums 2, 3, 5, 7, 11 have 1 + 2 + 4 + 6 + 2 = 15 ways, so P = 15/36 = 5/12.

4. Playing Cards

A standard deck has 52 cards: 4 suits of 13 cards each.

• Red suits: Hearts and Diamonds (26 cards). Black suits: Spades and Clubs (26 cards).
• Each suit has Ace, 2 to 10, Jack, Queen and King.
• Face cards (Jack, Queen, King): 12 in total. If Aces are counted as honours, there are 16 honour cards.
• 4 Aces, 4 Kings, 4 Queens, 4 Jacks.

Example: P(drawing a king) = 4/52 = 1/13. P(a red face card) = 6/52 = 3/26.

Example: P(a king or a heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 (subtract the king of hearts counted twice).

Example: Two cards are drawn together. P(both are aces) = 4C2 ÷ 52C2 = 6 ÷ 1326 = 1/221.

5. Balls in a Bag

These questions use combinations because balls are drawn together (or without order).

Example: A bag has 5 red and 3 blue balls. Two balls are drawn at random. P(both red) = 5C2 ÷ 8C2 = 10/28 = 5/14. P(one red and one blue) = (5C1 × 3C1) ÷ 28 = 15/28.

With replacement vs without replacement: With replacement, the total stays the same each time (independent events). Without replacement, the total reduces after each draw.

Example: Same bag, two draws with replacement. P(both red) = 5/8 × 5/8 = 25/64. Without replacement: 5/8 × 4/7 = 5/14.

6. Addition and Multiplication Rules

• Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
• Multiplication rule: For independent events, P(A ∩ B) = P(A) × P(B).
• "At least one" trick: P(at least one) = 1 − P(none).

Example: A can hit a target with probability 1/3 and B with probability 1/4. P(target is hit) = 1 − (2/3 × 3/4) = 1 − 1/2 = 1/2.

Example: P(A solves a problem) = 1/2, P(B solves) = 1/3. P(exactly one solves) = 1/2 × 2/3 + 1/2 × 1/3 = 2/6 + 1/6 = 1/2.

7. Odds

• Odds in favour = favourable : unfavourable.
• Odds against = unfavourable : favourable.
• If odds in favour are a : b, then P(E) = a ÷ (a + b).

Example: Odds in favour of an event are 3 : 5. P(event) = 3/8 and P(not event) = 5/8.

8. Probability in CSAT: A Logical Approach

UPSC CSAT rarely asks formula-heavy probability. Instead, it frames questions around everyday logic, such as the chance that two people pick the same item, or how many draws guarantee a matching pair. The "worst case" idea is important: to be sure of getting two socks of the same colour from a drawer with 3 colours, you must take out 4 socks. This is a certainty question, not a probability calculation, but it appears under the same theme.

Common Mistakes to Avoid

1. Adding probabilities of events that are not mutually exclusive, without subtracting the overlap.
2. Forgetting to reduce the total in "without replacement" questions.
3. Counting HT and TH as one outcome when tossing two coins.
4. Confusing odds with probability.
5. Getting a probability greater than 1, which is always a sign of an error.

Preparation Strategy

• Learn the coin, dice and card facts until they are automatic.
• Practise 10 questions each on coins, dice, cards and balls.
• Use the complement method for every "at least" question.
• Revise combination formulas, as most bag and card questions need nCr.
• Solve previous year bank and SSC questions, then a few CSAT logic-based ones.

Practice Questions with Answers

1. Two dice are thrown. Find P(sum is 10 or more). Answer: 3 + 2 + 1 = 6 ways, so 6/36 = 1/6.
2. A card is drawn from a deck. Find P(neither a heart nor a king). Answer: 1 − 4/13 = 9/13.
3. A number is chosen from 1 to 20. Find P(it is a multiple of 3 or 5). Answer: 6 + 4 − 1 = 9, so 9/20.
4. Four coins are tossed. Find P(at least 3 heads). Answer: (4C3 + 4C4) ÷ 16 = 5/16.

Conditional Probability (Basic Idea)

Sometimes we know that one event has already happened, and we want the probability of another event given this information. This is called conditional probability, written P(A | B), and read as "probability of A given B". The formula is P(A | B) = P(A ∩ B) ÷ P(B).

Example: A die is thrown and the result is known to be even. What is the probability that it is greater than 3? Even outcomes are {2, 4, 6}. Of these, 4 and 6 are greater than 3. So the answer is 2/3. Notice that the sample space shrinks from 6 outcomes to 3 because of the given information.

Example: In a class, 60% of students play cricket, 40% play football and 25% play both. If a student plays football, the probability that the student also plays cricket = 25 ÷ 40 = 5/8.

Real-Life Uses of Probability

Probability is not just an exam topic. Weather departments use it to forecast rain, insurance companies use it to calculate premiums, doctors use it to judge the risk of disease, and election analysts use it in opinion polls. In public policy, probability helps governments plan for disasters such as floods and cyclones by estimating how often they may occur. Understanding these uses helps UPSC aspirants connect quantitative thinking with governance, which is useful in the Essay and General Studies papers as well.

When you read any probability question, first write the sample space in a short form, then mark the favourable outcomes. This one habit removes most careless errors and keeps you from mixing up "and" with "or".

Related Notes

• Permutation, Combination and Probability for CSAT
• Percentage Notes
• Ratio and Proportion Notes
• All Quantitative Aptitude Notes

Frequently Asked Questions

What is the basic formula of probability?

Probability of an event = number of favourable outcomes ÷ total number of possible outcomes.

What is the range of probability?

Probability always lies between 0 and 1. Zero means impossible and one means certain.

How many outcomes are there when two dice are thrown?

There are 36 outcomes, since each die has 6 faces and 6 × 6 = 36.

Which sum is most likely with two dice?

The sum 7 is most likely, with 6 favourable outcomes out of 36.

How many face cards are in a deck of 52 cards?

There are 12 face cards: 4 Jacks, 4 Queens and 4 Kings.

What is a complementary event?

It is the event that the given event does not happen. P(not E) = 1 − P(E).

What are mutually exclusive events?

Events that cannot occur at the same time, such as getting both a head and a tail on a single toss.

What are independent events?

Events where the result of one does not affect the other. For them, P(A and B) = P(A) × P(B).

What is the difference between odds and probability?

Probability compares favourable outcomes with all outcomes. Odds compare favourable outcomes with unfavourable outcomes.

How do you solve "at least one" probability questions?

Find the probability of none and subtract it from 1.

Is probability asked in UPSC CSAT?

Yes, occasionally, usually as a logic-based question. Bank and SSC exams ask it more directly.

What is the probability of getting a head when a coin is tossed?

For a fair coin, it is 1/2.