Introduction to Probability

Probability describes uncertainty on a scale from impossible to certain. The current syllabus combines theoretical and experimental probability, sample spaces, events, tables and tree diagrams.

Core ideas

A random experiment has an outcome that cannot be predicted with certainty before it is performed. The sample space lists all possible outcomes; an event is a set of outcomes.

For equally likely outcomes, P(E)=number of favourable outcomes ÷ total outcomes. Probability lies between 0 and 1: 0 means impossible, 1 means certain. The empirical probability after n trials is observed frequency of E ÷ n; it may vary from run to run and often becomes more stable with many trials.

A table or tree diagram helps list outcomes without missing combinations. For a sequence of independent events, multiply probabilities along a branch; add probabilities for separate favourable branches.

Worked examples

Theoretical probability

A fair die is rolled. Outcomes are {1,2,3,4,5,6}. For an even number, favourable outcomes are {2,4,6}, so P(even)=3/6=1/2.

Tree diagram reasoning

Toss a fair coin twice. The branches are H then H/T, and T then H/T, giving HH, HT, TH, TT. Exactly one head occurs in HT or TH, so the probability is 2/4=1/2.

Empirical probability

A spinner lands on blue 18 times in 60 spins. The estimated probability is 18/60=0.3. This is an experimental estimate, not proof that the theoretical probability is exactly 0.3.

Practice questions with answers

A bag contains 3 red and 5 blue counters. One counter is drawn at random. Find P(red).

There are 8 equally likely counters and 3 are red, so P(red)=3/8.

A fair coin is tossed and a fair die rolled. How many equally likely outcomes are there?

2×6=12. A table with two coin rows and six die columns displays all outcomes.

In 40 trials, an event happens 11 times. Find its empirical probability.

11/40=0.275.

Practical investigation

Roll a die 30 times and record the outcomes in a tally table. Estimate P(rolling 6), compare it with the theoretical value 1/6 and explain why the estimates may differ. If you can, combine class results and compare again.

State the experiment, number of trials, observed frequency and limitations of the sample.

These are original revision notes aligned to the CBSE Class IX 2026–27 curriculum. Use them with your prescribed NCERT book and your teacher’s directions. Official CBSE syllabus · Official NCERT textbook resource