Randomness and the probability scale
A random experiment is one where we know all the things that could happen, but we cannot say for sure which one will happen next. Tossing a coin and rolling a die are random.
Probability measures how likely an event is, on a scale from 0 to 1:
- 0 = impossible (rolling 7 on a normal die)
- between 0 and ½ = unlikely
- ½ = even chance (heads on a fair coin)
- between ½ and 1 = likely
- 1 = certain
It can be written as a fraction, a decimal or a percentage: ½ = 0.5 = 50%.
Empirical (experimental) probability
Do an experiment many times (each time is a trial) and count how often the event happens.
Empirical probability P(E) = number of trials in which E happened ÷ total number of trials.
Example: 200 tosses, 108 heads → P(H) = 108 ÷ 200 = 0.54. The answer changes a little each time you repeat the experiment, but with more trials it settles near the true value.
We can use it to predict: if 2% of tested bulbs are faulty, about 2% of 5000 bulbs (100 bulbs) will be faulty.
Sample spaces and events
An outcome is one possible result. The sample space S is the list of all outcomes. An event is a group of outcomes we are interested in.
- One die: S = {1, 2, 3, 4, 5, 6}. Event "even" = {2, 4, 6}.
- Two coins: S = {HH, HT, TH, TT}.
When all outcomes are equally likely: P(E) = number of outcomes in E ÷ number of outcomes in S.
P(not E) = 1 − P(E). The probabilities of all outcomes add up to 1.
Tree diagrams and tables
When two things happen (two coins, two dice, two picks), list the outcomes neatly:
- Tree diagram: branches for the first stage, then branches from each for the second. Write the probability on each branch. Multiply along a path to get that outcome's probability; add the paths that belong to your event.
- Table (grid): first thing along the top, second down the side. Two dice give 6 × 6 = 36 cells. Count the cells you want.
Counting rule: if the first stage has m outcomes and the second has n, together there are m × n outcomes.
Board exam pattern
Statistics and Probability carries 10 marks in CBSE Class 9 (2026-27). Expect: place events on the probability scale, find empirical probability from a table of results, write a sample space, use a tree or a dice table, and MCQs such as "probability can never be…".
Try it at home
Toss a coin 20 times and write H or T each time. Find P(H). Now join your results with 4 friends' results (100 tosses). Is the answer closer to ½? Then roll two dice 36 times and tally the sums. Which sum came most often? Compare with the table: sum 7 should be the most common.
Key formulas and definitions
- 0 ≤ P(E) ≤ 1; P(impossible) = 0, P(certain) = 1
- Empirical P(E) = times E happened ÷ total trials
- Equally likely outcomes: P(E) = outcomes in E ÷ outcomes in S
- P(not E) = 1 − P(E)
- Tree: multiply along a path, add between paths
- m choices then n choices → m × n outcomes (2 coins: 4, 2 dice: 36)
Worked examples
1. Place on the probability scale: (a) picking a red ball from a bag of only red balls, (b) a newborn baby being a girl, (c) getting 0 on a normal die.
(a) certain → 1. (b) about even chance → ½. (c) impossible → 0.
2. A coin is tossed 200 times and shows heads 108 times. Find the empirical probability of heads and of tails.
P(H) = 108 ÷ 200 = 0.54. Tails came 92 times, so P(T) = 92 ÷ 200 = 0.46. Check: 0.54 + 0.46 = 1.
3. Out of 1000 bulbs tested, 20 were faulty. Find the probability that a bulb is faulty. About how many faulty bulbs are expected in 5000?
P(faulty) = 20 ÷ 1000 = 0.02. Expected = 0.02 × 5000 = 100 bulbs.
4. A die is rolled. Write the sample space and find P(even) and P(not a 6).
S = {1, 2, 3, 4, 5, 6}. Even = {2, 4, 6} → 3/6 = ½. P(6) = 1/6, so P(not 6) = 1 − 1/6 = 5/6.
5. Two coins are tossed. Use a tree diagram to find P(at least one head).
Branches: H or T, then H or T → HH, HT, TH, TT, each ½ × ½ = ¼. At least one head: HH, HT, TH → ¼ + ¼ + ¼ = 3/4. Or: 1 − P(TT) = 1 − ¼ = 3/4.
6. Two dice are rolled. Find P(sum is 7) and P(sum is 10 or more).
36 outcomes. Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6/36 = 1/6. Sum ≥ 10: (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) → 6/36 = 1/6.
7. A bag has 3 red and 2 blue balls. A ball is picked, put back, and a second ball is picked. Find P(both red).
P(red) = 3/5 each time (the ball is put back). Tree path red → red: 3/5 × 3/5 = 9/25 = 0.36.
Common mistakes
- Giving a probability greater than 1 or less than 0. It must be between 0 and 1.
- Treating HT and TH as the same outcome. With two coins there are 4 outcomes, not 3.
- Thinking empirical probability is always equal to the theoretical value. It only comes close with many trials.
- Adding along a tree path instead of multiplying. Multiply along a path; add different paths.