Probability Class 10 Notes: Formula, Examples & Questions

Probability is Chapter 14 of NCERT Class 10 Maths. It is one of the shortest and most scoring chapters in the book: almost every board paper has a question on coins, dice, playing cards or balls in a bag. If you know the sample spaces well and follow one simple formula, you can get full marks here.

These notes cover every idea in the chapter, the sample spaces you must know by heart, solved examples and the common mistakes that cost marks.

Chapter at a glance

Probability tells us how likely an event is, as a number from 0 to 1.
Theoretical probability: P(E) = number of favourable outcomes ÷ total number of possible outcomes.
P(E) + P(not E) = 1.
A sure event has probability 1. An impossible event has probability 0.
Most board questions come from coins, dice, a deck of 52 cards and balls or numbered cards drawn from a bag.

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Basic terms

Experiment: an action whose result is uncertain, like tossing a coin or rolling a die.
Outcome: one possible result. Getting a 4 on a die is an outcome.
Sample space: the list of all possible outcomes. For a die, it is {1, 2, 3, 4, 5, 6}.
Event: a set of outcomes we are interested in, like “getting an even number”.
Equally likely outcomes: each outcome has the same chance. A fair coin or a fair die gives equally likely outcomes. The theoretical formula works only for such outcomes.
Elementary event: an event with only one outcome. The sum of the probabilities of all elementary events of an experiment is 1.

The probability formula

For an event E:

P(E) = Number of outcomes favourable to E ÷ Total number of possible outcomes

Three facts follow from this:

0 ≤ P(E) ≤ 1. If your answer is more than 1 or negative, you have made a mistake.
Sure (certain) event: P = 1. Example: getting a number less than 7 on a die.
Impossible event: P = 0. Example: getting 8 on a die.

Complementary events

The event “not E” is written as Ē (E bar) and is called the complement of E. Since one of the two must happen:

P(E) + P(Ē) = 1, so P(Ē) = 1 − P(E)

This saves time. If the probability that it rains is 0.3, the probability that it does not rain is 0.7.

Experimental vs theoretical probability

You met experimental (empirical) probability in Class 9. It is based on trials actually done: P = number of trials in which the event happened ÷ total number of trials. Theoretical (classical) probability is calculated using the formula above, assuming equally likely outcomes. When an experiment is repeated a very large number of times, the experimental value comes close to the theoretical value.

Sample spaces you must know

Coins

Coins tossed	Sample space	Total outcomes
One coin	H, T	2
Two coins	HH, HT, TH, TT	4
Three coins	HHH, HHT, HTH, THH, HTT, THT, TTH, TTT	8

For n coins, total outcomes = 2ⁿ. Note that HT and TH are different outcomes.

Dice

One die has 6 outcomes. Two dice thrown together have 6 × 6 = 36 outcomes, written as pairs (1, 1), (1, 2) … (6, 6). The most useful table is the number of ways to get each sum:

Sum	2	3	4	5	6	7	8	9	10	11	12
Ways	1	2	3	4	5	6	5	4	3	2	1

Sum 7 is the most likely (6 ways). A doublet means both dice show the same number: there are 6 doublets.

A deck of 52 playing cards

4 suits of 13 cards each: spades ♠ and clubs ♣ (black), hearts ♥ and diamonds ♦ (red).
26 red cards and 26 black cards.
Each suit has A, 2 to 10, J, Q, K.
Face cards: jack, queen and king only, so 3 × 4 = 12 face cards. The ace is not a face card.
4 aces, 4 kings, 4 queens, 4 jacks.

Solved examples

Example 1. A die is thrown once. Find the probability of getting a prime number.
Prime numbers on a die: 2, 3, 5. Favourable = 3, total = 6. P = 3/6 = 1/2.

Example 2. Two dice are thrown together. Find the probability that the sum is 8.
Favourable pairs: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) = 5. Total = 36. P = 5/36.

Example 3. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a red face card, (ii) not an ace.
(i) Red face cards = J, Q, K of hearts and diamonds = 6. P = 6/52 = 3/26.
(ii) P(ace) = 4/52 = 1/13, so P(not an ace) = 1 − 1/13 = 12/13.

Example 4. Two coins are tossed together. Find the probability of getting at least one head.
Outcomes: HH, HT, TH, TT. At least one head: HH, HT, TH = 3. P = 3/4. (Shortcut: 1 − P(no head) = 1 − 1/4.)

Example 5. A bag contains 3 red and 5 black balls. One ball is drawn at random. Find P(red) and P(not red).
Total = 8. P(red) = 3/8. P(not red) = 1 − 3/8 = 5/8.

Example 6. What is the probability that a leap year has 53 Sundays?
A leap year has 366 days = 52 weeks + 2 extra days. The 2 extra days can be: (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun) = 7 cases. Sunday appears in 2 of them. P = 2/7.

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Common mistakes to avoid

Counting HT and TH as one outcome. With two coins, there are 4 outcomes, not 3.
Using 21 instead of 36 for two dice. (2, 5) and (5, 2) are different outcomes.
Counting the ace as a face card. There are only 12 face cards.
Not simplifying the fraction. Write 6/52 as 3/26.
Missing the “not” in the question. Read carefully: “not a king”, “at least one”, “at most one”.
Missing numbers in a range. In “a number from 1 to 30 divisible by 3”, list them: 3, 6 … 30 = 10 numbers.

Important questions

A die is thrown once. Find the probability of getting a number (i) greater than 4, (ii) less than or equal to 4. (2 marks)
Two dice are thrown at the same time. Find the probability that the sum of the numbers is (i) 7, (ii) a prime number. (3 marks)
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a red king, (ii) a face card, (iii) a spade. (3 marks)
A box contains 5 red, 8 white and 4 green marbles. One marble is taken out at random. Find the probability that it is (i) red, (ii) white, (iii) not green. (3 marks)
If P(E) = 0.05, what is the probability of “not E”? (1 mark)
Which of these cannot be the probability of an event: 2/3, −1.5, 15%, 0.7? Give a reason. (1 mark)

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FAQs

Which chapter is Probability in Class 10 Maths?

In the current (rationalised) NCERT book, Probability is Chapter 14, the last chapter. Older books and some websites still call it Chapter 15.

Can probability be more than 1 or negative?

No. The probability of any event is always between 0 and 1, both included. A negative value or a value above 1 means there is a counting error.

What is the difference between Probability and Statistics in Class 10?

Statistics deals with the mean, median and mode of data that has already been collected. Probability tells how likely a future outcome is. Both chapters are scoring and are usually revised together.

Is Probability important for the board exam?

Yes. It usually appears as MCQs, a short-answer question or both, and the questions are direct. It is one of the easiest chapters in which to score full marks.

More Class 10 Maths notes

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