Class 10 Probability Complete Guide 2027 — Cards, Dice, Coins Solved Problems + Shortcuts

In short: Probability carries 4 marks in CBSE Class 10 Maths 2027 and is arguably the easiest chapter on the paper. One formula — P(E) = favourable outcomes / total outcomes — does all the work. The challenge is not the maths but the counting: knowing there are 52 cards in a deck, 6 outcomes on a die, and how to count favourable cases correctly. Master the card-deck facts below and these are the freest 4 marks on your paper.

Probability is Chapter 14 in CBSE Class 10 Maths — the final chapter of the syllabus. It sits under the Statistics and Probability unit (11 marks total, shared with Statistics). Probability alone carries about 4 marks, usually one 2-mark and one 2-mark question, or sometimes one 4-mark question.

This is the chapter where students either score full marks effortlessly or throw away marks on counting errors. The formula is trivial. The skill is in correctly identifying the sample space and favourable outcomes. For marks distribution across all chapters, see the CBSE Class 10 chapter-wise weightage.

The One Formula You Need

P(E) = Number of favourable outcomes / Total number of outcomes

That is it. Every problem in this chapter uses this single formula.

Complementary event: P(not E) = 1 − P(E)

This saves time. If finding P(E) directly is hard, find P(not E) and subtract from 1.

Key facts:

• 0 ≤ P(E) ≤ 1 (probability is always between 0 and 1)
• P(impossible event) = 0
• P(certain event) = 1
• P(E) + P(not E) = 1

The Card Deck — Memorise This

A standard deck of 52 playing cards appears in nearly every board paper. Memorise these counts:

Memory trick: "4 suits, 13 each, face = JQK." Aces are NOT face cards. This is the most common trap in board exams — students count aces as face cards and get 16 instead of 12.

Common trick question counts:

• "Neither a king nor a queen" = 52 − 4 − 4 = 44
• "A red king" = 2 (king of hearts + king of diamonds)
• "A face card that is also red" = 6

Dice and Coins — Quick Reference

One Die (6 outcomes: 1, 2, 3, 4, 5, 6)

• P(even number) = 3/6 = 1/2 (outcomes: 2, 4, 6)
• P(prime number) = 3/6 = 1/2 (outcomes: 2, 3, 5 — note: 1 is NOT prime)
• P(number > 4) = 2/6 = 1/3 (outcomes: 5, 6)
• P(perfect square) = 2/6 = 1/3 (outcomes: 1, 4)

Two Dice (36 outcomes)

When two dice are thrown, total outcomes = 6 × 6 = 36.

• P(sum = 7) = 6/36 = 1/6 (outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1))
• P(doublet) = 6/36 = 1/6 (outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6))
• P(sum = 12) = 1/36 (only (6,6))

Key trap: 1 is NOT a prime number. Board exams test this every year. Prime numbers on a die are 2, 3, 5 — that is 3 outcomes, not 4.

One Coin (2 outcomes: H, T)

Two Coins (4 outcomes: HH, HT, TH, TT)

Three Coins (8 outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT)

Pattern: n coins = 2ⁿ total outcomes.

Board Exam Question Types — Solved

Type 1: Cards (Most Common — 2-4 marks)

"A card is drawn at random from a well-shuffled deck. Find the probability of getting (i) a red face card (ii) neither an ace nor a king."

(i) Red face cards = 6 (J, Q, K of hearts + J, Q, K of diamonds) P(red face card) = 6/52 = 3/26

(ii) Aces + Kings = 4 + 4 = 8. Cards that are neither = 52 − 8 = 44 P(neither ace nor king) = 44/52 = 11/13

Type 2: Dice (2-3 marks)

"Two dice are thrown simultaneously. Find the probability that the sum of numbers is (i) 8 (ii) at least 10."

Total outcomes = 36

(i) Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 outcomes P(sum = 8) = 5/36

(ii) Sum ≥ 10: sum=10: (4,6)(5,5)(6,4)=3; sum=11: (5,6)(6,5)=2; sum=12: (6,6)=1. Total = 6 P(sum ≥ 10) = 6/36 = 1/6

Type 3: Numbered Items (2-3 marks)

"A box contains 5 red balls, 8 white balls, and 4 green balls. A ball is drawn at random. Find the probability that the ball is (i) red (ii) not green."

Total = 5 + 8 + 4 = 17

(i) P(red) = 5/17 (ii) P(not green) = 1 − P(green) = 1 − 4/17 = 13/17

Type 4: Complementary Probability (1-2 marks)

"The probability of getting a defective bulb from a lot is 0.06. Find the probability of getting a non-defective bulb."

P(non-defective) = 1 − 0.06 = 0.94

Common Mistakes That Cost Marks

1. Counting aces as face cards. Face cards = Jacks + Queens + Kings = 12. Aces are NOT face cards. The count is 12, not 16. This single mistake has cost lakhs of students marks over the years.

2. Counting 1 as a prime number. 1 is neither prime nor composite. Prime numbers on a die: 2, 3, 5. That is 3 outcomes, not 4.

3. Wrong total outcomes for two dice. Two dice have 6 × 6 = 36 outcomes, not 12. Each die is independent, so the outcomes multiply.

4. Not reducing fractions. Board examiners expect simplified fractions. Write 3/26, not 6/52. Write 1/6, not 6/36. Not simplifying may cost half a mark.

5. Confusing "at least" with "exactly." "At least 2 heads" means 2 or 3 heads (with 3 coins). "Exactly 2 heads" means only 2 heads. The word "at least" means ≥, "at most" means ≤.

4-Week Practice Plan

Week 1: Single Events — Coins, Dice, Cards

• Read NCERT Section 14.1 and 14.2
• Memorise the card deck table above — test yourself daily
• Solve 15 single-die and single-coin problems
• Solve 10 card problems from NCERT exercises

Week 2: Two Dice and Multiple Events

• List all 36 outcomes for two dice in a 6×6 grid — this visual helps enormously
• Solve all two-dice problems from NCERT
• Practice "at least," "at most," and "exactly" type questions — 5 of each
• Solve NCERT Exercise 14.1 completely

Week 3: Mixed Problems and Complementary Events

• Solve numbered-ball, lottery ticket, and calendar problems
• Practice using P(not E) = 1 − P(E) — at least 5 problems
• Attempt CBSE Class 10 Maths important questions for this chapter
• Check: can you solve any card problem in under 2 minutes?

Week 4: Board Paper Practice

• Solve Probability questions from the last 5 years of CBSE board papers
• Time yourself: 2-mark questions in 2-3 minutes, 4-mark questions in 5 minutes
• Solve the CBSE 2027 sample paper Probability questions
• Final drill: 10 random card/dice problems in 15 minutes

Quick Revision Card for Exam Day

• P(E) = Favourable / Total
• P(not E) = 1 − P(E)
• 0 ≤ P(E) ≤ 1 always
• Cards: 52 total, 4 suits, 13 each, 26 red, 26 black
• Face cards = 12 (J+Q+K only). ACES ARE NOT FACE CARDS.
• One die: 6 outcomes. Two dice: 36 outcomes.
• 1 is NOT prime. Primes on a die: 2, 3, 5
• "At least" = ≥, "At most" = ≤
• Always simplify fractions
• n coins → 2ⁿ outcomes

For instant practice on probability problems with all card, dice, and coin variations, try Padhaao's AI tutor — it generates fresh problems and tells you exactly where your counting went wrong.

Frequently Asked Questions

How many marks does Probability carry in CBSE Class 10 Maths 2027?

Probability carries approximately 4 marks in the CBSE Class 10 board exam 2027. It is part of the Statistics and Probability unit (11 marks total). Typically you will see one or two questions — either two 2-mark questions or one 4-mark question. The questions are almost always about cards, dice, or coins, so the patterns are very predictable.

Are aces considered face cards in a deck of 52 cards?

No. Aces are NOT face cards. Face cards are only Jacks, Queens, and Kings — 12 cards total (3 per suit × 4 suits). This is one of the most common mistakes in board exams. If a question asks for the probability of drawing a face card, the favourable outcomes are 12, not 16.

How many outcomes are there when two dice are thrown together?

When two dice are thrown simultaneously, there are 36 total outcomes (6 × 6). Each die can show any number from 1 to 6 independently, giving 36 combinations: (1,1), (1,2), ..., (6,6). A useful technique is to draw a 6×6 grid and list all pairs — this makes counting favourable outcomes for any sum or condition straightforward.

What is the difference between "at least" and "at most" in probability?

"At least n" means n or more (≥ n). "At most n" means n or fewer (≤ n). For example, with three coins: "at least 2 heads" includes the outcomes HHT, HTH, THH, HHH (4 outcomes). "At most 2 heads" includes TTT, HTT, THT, TTH, HHT, HTH, THH (7 outcomes). Reading these words carefully is essential to counting correctly.

Is Probability easy enough to score full marks in CBSE Class 10?

Yes, Probability is widely considered the easiest chapter in Class 10 Maths. The formula is simple and the question types are repetitive. The only risks are counting errors (wrong number of face cards, forgetting that 1 is not prime) and not simplifying fractions. If you memorise the card deck facts and practise 20-30 problems, full marks are very achievable. Many toppers consider this chapter their 4 "free" marks.

Probability is the chapter where careful counting beats mathematical ability. Memorise your card deck, remember that aces are not face cards and 1 is not prime, simplify your fractions, and those 4 marks are the easiest on the paper.