Probability: Full Chapter Notes for Class 10 Boards

Probability is the shortest chapter in the Class 10 Maths book and one of the most dependable sources of marks. Almost every question uses a single idea, favourable outcomes divided by total outcomes, and the difficulty is nearly always in counting those outcomes correctly. Once you can count with care, the chapter is easy.

This guide covers the definitions, the formula, the complement rule, and the standard experiments with coins, dice and cards, with worked examples and the slips to avoid. It is the notes version of the chapter. For worked board-style questions, continue to probability important questions and solved examples, and for the whole-chapter overview, see these notes on probability and this introduction to probability and statistics.

The Basic Language

The word fair matters. A fair coin and a fair die are not biased towards any outcome, and without that assumption, the formula below does not apply.

The Formula

For an experiment with equally likely outcomes, the theoretical probability of an event E is

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

Example. A die is thrown once. What is the probability of getting a number greater than 4?

The favourable outcomes are 5 and 6, which is 2 outcomes. The total is 6. So P = 2/6 = 1/3.

What the Number Tells You

• The probability of any event lies between 0 and 1, written 0 ≤ P(E) ≤ 1. A negative probability, or one above 1, signals a mistake.
• An event that is certain has probability 1. For example, getting a number less than 7 on a die.
• An event that is impossible has probability 0. For example, getting 8 on a die.
• The sum of the probabilities of all the outcomes of an experiment is 1.

The Complement Rule

The event "not E", written Ē, is the complement of E. Together, E and not E cover every possibility, so

P(E) + P(not E) = 1, which gives P(not E) = 1 − P(E).

This is the most useful shortcut in the chapter. When a question asks for "at least one", it is usually easier to find the chance of "none" and subtract from 1.

Example. A coin is tossed three times. What is the probability of getting at least one head?

The total number of outcomes is 2 × 2 × 2 = 8. The only outcome with no heads is TTT. So P(no head) = 1/8, and P(at least one head) = 1 − 1/8 = 7/8.

Coins: Counting Outcomes

For two coins, the probability of two heads is 1/4. The probability of exactly one head is 2/4 = 1/2, because both HT and TH count.

Dice: Counting Outcomes

One die has 6 outcomes. Two dice have 6 × 6 = 36 outcomes, not 12, and this is the most frequent counting mistake in the chapter. The best habit is to write the sums in a table.

A sum of 7 is the most likely, because it can be made in 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). The table adds up to 36/36 = 1, which is a quick way to check yourself.

Example. Two dice are thrown together. What is the probability that the product of the numbers is 12?

The pairs are (2,6), (3,4), (4,3), (6,2). That is 4 outcomes, so P = 4/36 = 1/9.

A Pack of Cards

A standard pack has 52 cards, in 4 suits of 13 cards each.

Example. One card is drawn at random from a well-shuffled pack. What is the probability that it is a red face card?

There are 2 red suits with 3 face cards each, so 6 red face cards. P = 6/52 = 3/26.

Balls in a Bag and Spinners

Many questions use a bag with different coloured balls, or a spinner with numbered sectors. The method is identical. Count the total, count the favourable, and divide.

Example. A bag has 5 red, 8 white and 7 green balls. One ball is drawn at random. What is the probability that it is not red?

Total = 20. Red = 5, so not red = 15. P = 15/20 = 3/4. Alternatively, P(red) = 5/20 = 1/4, so P(not red) = 1 − 1/4 = 3/4.

A Calendar Question That Repeats

A non-leap year has 365 days, which is 52 weeks and 1 extra day. That extra day can be any of the 7 days of the week, so the probability of 53 Sundays is 1/7. In a leap year, there are 2 extra days, and the pairs that include a Sunday are two out of seven, so the probability of 53 Sundays is 2/7.

Experimental and Theoretical Probability

Probability can also be found by experiment: if you toss a coin 100 times and get 54 heads, the experimental probability of heads is 54/100. Theoretical probability, from the formula, is 1/2. The two are close, and the more times you repeat the experiment, the closer they get. The Class 10 chapter works with theoretical probability, but the connection helps you understand why the formula is a good prediction.

Slips That Cost Marks

• Counting two dice as 12 outcomes. It is 36.
• Forgetting that the fraction must be simplified, for example leaving 6/36 instead of 1/6.
• Counting HT and TH as one outcome. They are different outcomes.
• Writing a probability greater than 1, or giving it as a negative number.
• Using the formula when outcomes are not equally likely, for example treating "rain or no rain" as 1/2.
• Ignoring the complement, and doing a long count for "at least one".

Probability in Your Textbooks

Probability is the last chapter of the Class 10 Maths book in both the CBSE and the Karnataka SSLC textbooks, and it is retained in both, according to published listings. Revise with the chapter MCQs and the important questions. In Class 11, probability returns with more formal ideas such as sample spaces and the axioms, as in these notes on probability in Class 11, which feed into JEE Main and KCET mathematics. Confirm the current portions with your own board.

Frequently Asked Questions

Q1. What is the formula for probability?

For equally likely outcomes, P(E) = (number of favourable outcomes) ÷ (total number of possible outcomes).

Q2. What is the range of probability?

The probability of any event lies between 0 and 1, including both. A certain event has probability 1 and an impossible event has probability 0.

Q3. What is the probability of getting a sum of 7 with two dice?

There are 6 favourable outcomes out of 36, so the probability is 6/36 = 1/6.

Q4. How many cards are there in a pack, and how many face cards?

A pack has 52 cards, of which 12 are face cards: the jack, queen and king of each of the four suits.

Q5. What is the complement rule?

The probability that an event does not happen equals 1 minus the probability that it does: P(not E) = 1 − P(E).

Count First, Then Divide

Probability questions reward patience at the counting stage and speed at the arithmetic stage. List the outcomes, check that they are equally likely, count the favourable ones, and divide. If the question says "at least", try the complement.