Probability Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
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PDF Notes: Probability Notes for SSC Exams
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Probability tells us how likely an event is to happen. It is a scoring topic in SSC CGL, CHSL and MTS Quantitative Aptitude, because most questions are based on a few coins, dice, cards or balls and use simple formulas. These probability notes for SSC explain every formula with solved examples on coins, dice, cards and balls, followed by practice questions in simple language.
Basic Terms
| Term | Meaning | Example |
|---|---|---|
| Experiment | An action with different possible results | Tossing a coin |
| Sample space (S) | The set of all possible outcomes | {H, T} for one coin |
| Event (E) | A set of outcomes that we want | Getting a head |
| Favourable outcomes | Outcomes that make the event happen | 1 head out of 2 |
| Sure event | An event that always happens; probability = 1 | Getting a number from 1 to 6 on a die |
| Impossible event | An event that can never happen; probability = 0 | Getting 7 on a die |
Main Formulas
| Rule | Formula |
|---|---|
| Probability of an event | P(E) = Number of favourable outcomes / Total number of outcomes |
| Range | 0 ≤ P(E) ≤ 1 |
| Complement | P(not E) = 1 − P(E) |
| Addition rule | P(A or B) = P(A) + P(B) − P(A and B) |
| Mutually exclusive events | P(A or B) = P(A) + P(B) |
| Independent events | P(A and B) = P(A) × P(B) |
| Conditional probability | P(A | B) = P(A and B) / P(B) |
Coins and Dice: Useful Facts
- One coin has 2 outcomes, two coins have 4, and n coins have 2n outcomes.
- One die has 6 outcomes, and two dice have 36 outcomes.
- A pack of cards has 52 cards: 26 red and 26 black, in four suits (spades, clubs, hearts, diamonds) of 13 cards each.
- There are 12 face cards (king, queen and jack of each suit), 4 kings, 4 queens, 4 jacks and 4 aces.
Sum of Two Dice
| Sum | Number of ways | Probability |
|---|---|---|
| 2 or 12 | 1 | 1/36 |
| 3 or 11 | 2 | 2/36 |
| 4 or 10 | 3 | 3/36 |
| 5 or 9 | 4 | 4/36 |
| 6 or 8 | 5 | 5/36 |
| 7 | 6 | 6/36 = 1/6 |
Solved Examples
Example 1. A die is thrown once. Find the probability of getting a number greater than 4.
Favourable outcomes are 5 and 6, so P = 2/6 = 1/3.
Example 2. Three coins are tossed. Find the probability of getting at least one head.
P(no head) = 1/8, so P(at least one head) = 1 − 1/8 = 7/8.
Example 3. A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart.
P(king) = 4/52, P(heart) = 13/52, P(king of hearts) = 1/52. So P = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
Example 4. A bag has 5 red and 3 blue balls. Two balls are drawn one by one without replacement. Find the probability that both are red.
P = (5/8) × (4/7) = 20/56 = 5/14.
Example 5. Two dice are thrown. Find the probability that the sum is 7.
There are 6 favourable outcomes out of 36, so P = 1/6.
With and Without Replacement
| Type | Meaning | Effect |
|---|---|---|
| With replacement | The item is put back after drawing | Total number of items remains the same; events are independent |
| Without replacement | The item is not put back | Total number of items reduces; events are dependent |
Odds
- Odds in favour of an event = favourable outcomes : unfavourable outcomes.
- Odds against an event = unfavourable outcomes : favourable outcomes.
- If P(E) = 2/5, the odds in favour are 2 : 3.
Quick Tips
- Count the total outcomes first, then the favourable outcomes, and simplify the fraction at the end.
- Use the complement rule for "at least one" questions, because it saves time.
- For two dice, write the 36 outcomes in a table if you are unsure.
- Check that your answer lies between 0 and 1.
Quick Revision Points
- P(E) = favourable outcomes / total outcomes.
- P(not E) = 1 − P(E).
- For independent events, P(A and B) = P(A) × P(B).
- Two dice have 36 outcomes and three coins have 8 outcomes.
- A pack has 52 cards, with 12 face cards and 4 aces.
Practice Questions With Answers
- What is the probability of getting a head on a toss of one coin?
Answer: 1/2 - What is the probability of getting an even number on a die?
Answer: 1/2 - Two coins are tossed. Find the probability of getting two heads.
Answer: 1/4 - A card is drawn from a pack. Find the probability that it is a red card.
Answer: 1/2 - A card is drawn from a pack. Find the probability that it is an ace.
Answer: 1/13 - Two dice are thrown. Find the probability of a sum of 9.
Answer: 1/9 (4/36) - A bag has 4 white and 6 black balls. One ball is drawn. Find the probability that it is white.
Answer: 2/5 - Three coins are tossed. Find the probability of exactly two heads.
Answer: 3/8 - If P(E) = 0.35, find P(not E).
Answer: 0.65 - Two dice are thrown. Find the probability that both numbers are the same.
Answer: 1/6 (6/36)
How to Practise This Chapter
Solve five probability questions daily, two each on dice and cards and one on balls. Then attempt timed sets from our practice papers and the previous year papers page. Read the next chapter, Permutation and Combination, and see all topics on the Quantitative Aptitude notes page. For related statistics, read the Statistics notes.
Frequently Asked Questions
What is probability?
Probability is a number between 0 and 1 that shows how likely an event is to happen.
What is the formula of probability?
Probability is the number of favourable outcomes divided by the total number of outcomes.
What is the probability of a sure event and an impossible event?
The probability of a sure event is 1, and the probability of an impossible event is 0.
How many outcomes are there when two dice are thrown?
There are 6 times 6, which is 36 outcomes.
How many cards are there in a pack and how many are face cards?
A pack has 52 cards, and there are 12 face cards: the king, queen and jack of each suit.
What is the addition rule of probability?
P(A or B) = P(A) + P(B) minus P(A and B). For mutually exclusive events the last term is zero.
What are independent events?
Two events are independent if the happening of one does not change the probability of the other. For them P(A and B) = P(A) into P(B).
What is conditional probability?
Conditional probability is the probability of an event A when we already know that the event B has happened.
How do I solve "at least one" questions?
Find the probability of "none" and subtract it from 1.
What is the difference between drawing with and without replacement?
With replacement the total does not change, but without replacement the total reduces after each draw, so the events are dependent.
What are odds?
Odds in favour are the ratio of favourable outcomes to unfavourable outcomes of an event.
How many questions come from probability in SSC exams?
Usually one or two questions come from probability in SSC Quantitative Aptitude papers, but the number changes from year to year.
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Data Interpretation Notes for SSC Exams
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Permutation and Combination Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.