In this guide
Probability questions are counting questions with a division at the end. Probability = favourable outcomes ÷ total outcomes, so the whole skill is counting both correctly. The counting uses a few tools: listing small cases, multiplying independent choices, and the combination nCr when order does not matter. With those, dice, balls, cards and committees all fall into place.
The tools and why they work
Probability basics
- P(A) = favourable ÷ total, when all outcomes are equally likely.
- P(not A) = 1 − P(A). Either A happens or it does not, and the two add to 1.
- P(at least one) = 1 − P(none). "At least one" has many cases; "none" has one. Counting the one case and subtracting is faster.
- For independent events, P(A and B) = P(A) × P(B).
- For "A or B", add and subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
Counting
- Multiplication principle. If one choice can be made in m ways and a second in n ways, both can be made in m × n ways.
- Arrangements (order matters). n different items can be arranged in n! ways. Choosing and arranging r of n: nPr = n! ÷ (n − r)!.
- Selections (order does not matter). nCr = n! ÷ (r! × (n − r)!). For example, 8C3 = (8 × 7 × 6) ÷ (3 × 2 × 1) = 56.
- Repeated items. If a word has a letter repeated p times, divide n! by p!, because swapping the identical letters gives no new arrangement.
- Items that must stay together. Treat the group as one block, arrange the blocks, then multiply by the arrangements inside the block.
Standard sample spaces
| Experiment | Total outcomes | Worth remembering |
|---|---|---|
| One coin | 2 | |
| Three coins | 8 | Only one outcome has no tail |
| One die | 6 | |
| Two dice | 36 | Six doublets; a total of 7 has the most ways (6) |
| One card from 52 | 52 | 4 suits of 13; 26 red; 12 face cards (J, Q, K); 4 aces |
| Two balls from 10 | 10C2 = 45 | Use combinations when drawn together |
Worked examples
Example 1. Two dice are thrown. What is the probability of a total of 9?
Favourable: (3,6), (4,5), (5,4), (6,3). That is 4 of 36.
1/9.
Example 2. A bag has 6 red and 4 blue balls. Two are drawn together. What is the probability that both are blue?
Ways to pick 2 blue = 4C2 = 6. Ways to pick any 2 = 10C2 = 45.
6 ÷ 45 = 2/15.
Example 3. In how many ways can the letters of POLICY be arranged?
All six letters differ: 6! = 720.
Example 4. A committee of 3 men and 2 women is to be chosen from 6 men and 4 women. In how many ways?
Men: 6C3 = 20. Women: 4C2 = 6. Both choices are made, so multiply.
20 × 6 = 120.
Example 5. In how many ways can the letters of PREMIUM be arranged?
Seven letters, with M twice. 7! ÷ 2! = 5,040 ÷ 2 = 2,520.
Example 6. A die is thrown twice. What is the probability of at least one six?
P(no six) = 5/6 × 5/6 = 25/36.
P(at least one six) = 1 − 25/36 = 11/36.
Example 7. A bag has 5 red, 3 green and 2 white balls. Three are drawn together. What is the probability that exactly two are red?
Two red from 5: 5C2 = 10. One non-red from 5: 5C1 = 5. Favourable = 50.
Total = 10C3 = 120. Probability = 50/120 = 5/12.
Example 8. In how many arrangements of CLAIM do the vowels come together?
The vowels are A and I. Treat AI as one block: C, L, M and the block make 4 items, arranged in 4! = 24 ways.
Inside the block, A and I can swap: 2! = 2. Total = 48.
Choosing the right method
| Wording | Method |
|---|---|
| "Arranged", "in a row", "words formed" | Arrangements: n!, nPr |
| "Selected", "chosen", "committee", "drawn together" | Selections: nCr |
| "At least one" | 1 − P(none) |
| "Drawn one after another without replacement" | Multiply probabilities, reducing the total each time |
| "Must come together" | Block method |
| Repeated letters | Divide by the factorial of each repeat |
Common mistakes
- Forgetting that (4,5) and (5,4) are different outcomes with two dice.
- Adding probabilities for "and" events instead of multiplying.
- Counting a king of hearts twice in "a king or a heart".
- Not dividing by the repeats in words like PREMIUM.
Practice set
- Three coins are tossed. Probability of at least one tail? 7/8. Only HHH has no tail: 1 − 1/8.
- In how many ways can the letters of CLAIM be arranged? 120. 5! with all letters different.
- One card is drawn from 52. Probability of a face card? 3/13. 12 face cards: 12/52.
- Two dice are thrown. Probability of a doublet? 1/6. Six doublets out of 36.
- A committee of 3 is chosen from 5 men and 3 women. Probability that all are men? 5/28. 5C3 = 10; 8C3 = 56.
- In how many arrangements of AGENT do the vowels come together? 48. Block AE with G, N, T: 4! × 2!.
- One card is drawn from 52. Probability of a king or a heart? 4/13. 4 + 13 − 1 = 16; 16/52.
- Two cards are drawn one after another without replacement. Probability both are aces? 1/221. 4/52 × 3/51 = 12/2,652.
What to do next
- Write out the two-dice grid once, so totals and doublets are familiar.
- Practise ten nCr values by hand until 5C2, 6C3, 8C3 and 10C3 are instant.
- Keep probability as a lower-priority topic; secure percentage and table DI first.
- See where it fits in a quant plan for LIC AAO.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Life Insurance Corporation of India website .
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