A die is thrown once. Which of the following is an impossible event?
- (a)getting 6
- (b)getting 0
- (c)getting an odd number
- (d)getting a number less than 7
Probability Class 10: notes and important questions
Revision notes, 37 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.
- 37 questions
- 10 multiple choice, 3 assertion–reason, 8 very short answer, 8 short answer, 5 long answer, 3 case study
- About 8 hours to master
Statistics & Probability unit: 11 of 80 theory marks (Statistics and Probability).
Revision notes
Theoretical (classical) probability
If an experiment has \(n\) equally likely outcomes, and \(m\) of them are favourable to event \(E\), then
\(P(E)=\dfrac{\text{number of outcomes favourable to }E}{\text{total number of possible outcomes}}=\dfrac mn\).
- \(0\le P(E)\le1\). A sure event has probability 1 and an impossible event has probability 0.
- Complementary events: \(P(E)+P(\bar E)=1\), where \(\bar E\) is "not \(E\)".
- An event with a single outcome is an elementary event. The probabilities of all elementary events of an experiment add to 1.
Standard sample spaces
- One coin: {H, T}. Two coins: {HH, HT, TH, TT}. Three coins: 8 outcomes.
- One die: {1, 2, 3, 4, 5, 6}. Two dice: 36 ordered pairs \((a,b)\). Sum 7 occurs 6 times, the most of any sum.
- A pack of 52 cards: 4 suits (spades ♠ and clubs ♣ black; hearts ♥ and diamonds ♦ red), 13 cards each. Face cards are king, queen and jack: 12 in all. There are 4 aces.
Worked example 1
Two dice are thrown. Find the probability that the sum is 9. Favourable: (3,6), (4,5), (5,4), (6,3), so \(P=\frac{4}{36}=\frac19\).
Worked example 2
A card is drawn from a well-shuffled pack. Find \(P(\text{a black face card})\). There are 6 black face cards, so \(P=\frac{6}{52}=\frac{3}{26}\).
Worked example 3
A bag has 6 red balls and some blue balls. The probability of drawing a blue ball is \(\frac35\). How many blue balls are there?
\(\frac{b}{6+b}=\frac35\Rightarrow5b=18+3b\Rightarrow b=9\).
Common errors
- Treating "HT" and "TH" as one outcome, or listing (2,3) but not (3,2) for two dice.
- Forgetting that 1 is neither prime nor composite.
- Counting 16 face cards (the aces are not face cards).
- Not reducing the fraction, or giving a probability greater than 1.
Board-exam tips
- Write the total number of outcomes and list the favourable outcomes explicitly. Each step earns a mark.
- Use \(P(\text{not }E)=1-P(E)\) when "not", "at least one" or "neither" is asked.
- Only the classical (theoretical) approach is examined. The optional Exercise 14.2 has been removed from the rationalised book.
Topics in this chapter: Basic properties of probability · Probability with dice · Probability with coins · Probability with playing cards · Simple events · Complementary events.
Route to 95: four steps
Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.
Step 1
Secure the basics
You know that probabilities lie between 0 and 1, can use P(not E) = 1 − P(E), and can find simple probabilities.
Step 2
Board standard
You can list outcomes for coins, dice and cards correctly and solve the standard board questions on each.
Step 3
Full marks on long answers
You can write complete long answers and case studies, listing favourable outcomes and giving answers in lowest terms.
Step 4
95+ stretch (HOTS)
You can handle two-dice products and sums, modified card packs and 'find the number of balls' problems.
Practice questions
Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 5 of the 37 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.
The probability of a sure event is
- (a)0
- (b)1
- (c)\(\frac12\)
- (d)more than 1
In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.
Assertion (A): The probability of getting a number greater than 6 in a single throw of a die is 0.
Reason (R): The probability of an impossible event is 0.
- (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true but Reason (R) is false.
- (d)Assertion (A) is false but Reason (R) is true.
Where marks are lost in Probability
- Treating HT and TH (or (2, 3) and (3, 2)) as one outcome. Fix: write the full sample space (4 for two coins, 36 for two dice) before counting.
- Counting 16 face cards by including aces. Fix: face cards are J, Q, K only: 12 in a pack; aces are not face cards.
- Forgetting that 1 is neither prime nor composite. Fix: list the primes explicitly when counting.
- Not reducing the fraction, or giving a probability above 1. Fix: simplify every answer and check 0 ≤ P ≤ 1.
- Not writing the total number of outcomes and the favourable list. Fix: write 'Total outcomes = 36, favourable = {(4, 5), (5, 4), …} = 4'; each part is marked.
- Modified packs (cards removed): using 52 as the total. Fix: recount the total and each category after removal, on paper.
Probability in our sample papers
Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).
- Sample paper 1 (Board standard): questions 17, 18, 25
- Sample paper 2 (Board standard): questions 18, 38
- Sample paper 3 (Above board): questions 18, 25, 31
- Sample paper 4 (Above board): questions 18, 35, 38
- Sample paper 5 (95+ challenge): questions 18, 25, 31