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Probability questions in Class 10 are usually worth easy marks, and they are lost in a predictable way: the counting goes wrong before any division happens. Two coins become three outcomes, a pair of dice becomes 11 sums, or a pack with missing cards still has 52. This post sets out the rules, works through coins, dice and cards, and finishes with a short routine.
What does Class 10 probability cover?
The CBSE Mathematics curriculum for Class X (2026-27) lists the classical definition of probability and "simple problems on finding the probability of an event". Statistics and probability together carry 11 of the 80 theory marks. The NCERT Class 10 probability chapter, 2026-27 reprint, defines the theoretical or classical probability of an event E as:
P(E) = (number of outcomes favourable to E) / (number of all possible outcomes of the experiment)
The definition only works when every outcome is equally likely. NCERT's example is a bag with 4 red balls and 1 blue ball: "red" and "blue" are not equally likely, but each of the five balls is.
Three facts follow and come up often:
0 ≤ P(E) ≤ 1. An impossible event has probability 0, a sure event 1.
P(E) + P(not E) = 1, so P(not E) = 1 − P(E).
Any answer above 1 or below 0 means the counting went wrong.
How do you handle coin questions?
Example. Two coins are tossed together. Find the probability of exactly one head.
List the outcomes: HH, HT, TH, TT. That is 4 equally likely outcomes. Exactly one head happens in HT and TH. So P = 2/4 = 1/2.
The trap is to list the outcomes as "two heads, one head, no heads" and answer 1/3. Those three are not equally likely, because "one head" can happen in two ways. Always list outcomes coin by coin.
How do you handle dice questions?
One die has 6 outcomes. P(prime number) uses the primes 2, 3 and 5, so P = 3/6 = 1/2. P(a number greater than 6) = 0, an impossible event.
Two dice have 6 × 6 = 36 outcomes, written as pairs (first die, second die). Keep the order: (2, 6) and (6, 2) are different outcomes.
Example 1: a sum. P(sum is 8). The pairs are (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). P = 5/36.
The trap: the sums run from 2 to 12, which is 11 values, but they are not equally likely. A sum of 2 needs (1, 1) only, while a sum of 7 has 6 pairs. So P(sum 8) is not 1/11.
Example 2: a doublet. (1, 1), (2, 2) up to (6, 6) gives 6 outcomes, so P = 6/36 = 1/6.
Example 3: at least one six. Use the complement. "No six on either die" has 5 × 5 = 25 outcomes. P(at least one six) = 1 − 25/36 = 11/36. Counting directly also gives 11: six pairs with a six on the first die, six with a six on the second, minus (6, 6) counted twice.
Example 4: an even product. The product is odd only when both numbers are odd: 3 × 3 = 9 outcomes. P(even product) = 1 − 9/36 = 27/36 = 3/4.
How do you handle playing card questions?
NCERT describes the pack: 52 cards in 4 suits of 13 (spades, hearts, diamonds, clubs). Clubs and spades are black; hearts and diamonds are red. Each suit has ace, king, queen, jack and 10 down to 2. Kings, queens and jacks are face cards, so there are 12 face cards. The ace is not a face card.
Example 1. One card is drawn. P(red face card) = 6/52 = 3/26, since each red suit has 3 face cards.
Example 2. P(not an ace) = 1 − 4/52 = 48/52 = 12/13.
Example 3: a changed pack. The king, queen and jack of hearts are removed, and a card is drawn from the rest. Now there are 49 cards. P(heart) = 10/49, since 13 − 3 = 10 hearts remain. P(face card) = 9/49.
The CBSE sample question paper for 2026-27 uses this pattern, with some cards missing from the pack, as well as two-dice questions on products and on numbers less than 4.
What mistakes cost marks in probability?
Unequal outcomes in the list, as in the coin and dice traps above.
Not reducing the total when cards are removed.
Counting the ace as a face card.
Counting (2, 6) and (6, 2) once. With two dice, order matters.
Not simplifying. 27/36 should be written 3/4.
When you move to Class 11 and 12, the reasoning errors change shape. Our post on common probability mistakes in JEE maths covers independence and conditional probability.
A practice routine for probability
Day 1: Write the full 36-outcome grid for two dice once, by hand. Use it to answer five questions on sums.
Day 2: Card questions. Write the pack's structure (suits, colours, face cards) at the top of the page each time before answering.
Day 3: "At least one" and "not" questions, all by complement.
Day 4: Five mixed questions in 10 minutes. Next to each answer, check it lies between 0 and 1.
If a problem confuses you, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can see where your outcome list went wrong.
Before your next probability question, write the sample space in full before you count anything.
