What this chapter is about
Probability is the branch of mathematics that measures how likely an event is to happen. When you toss a coin, you cannot be certain whether it will land heads or tails, but you can say something sensible about the chances. This chapter introduces you to the language and basic calculations of probability using experiments you can actually perform.
At Class 9, you meet probability through what is called the experimental or empirical approach. Instead of assuming outcomes are equally likely, you actually conduct trials — tossing coins, rolling dice, drawing cards, recording observations — and use the results to estimate probabilities. This connects mathematics to real-world data and prepares you for the theoretical approach you will study later.
After working through this chapter, you should be able to describe random experiments, list possible outcomes, conduct repeated trials, record frequencies, and calculate the experimental probability of any event. You will also understand that as the number of trials increases, experimental probability tends to settle down to a stable value.
Key ideas
- A random experiment is an action whose outcome cannot be predicted with certainty, such as tossing a coin or rolling a die.
- An outcome is a single possible result of an experiment; the set of all outcomes is called the sample space.
- An event is any collection of one or more outcomes that we are interested in.
- Experimental probability of an event equals the number of trials in which the event occurred divided by the total number of trials.
- Probability always lies between 0 and 1 (inclusive). An impossible event has probability 0; a certain event has probability 1.
- As the number of trials increases, the experimental probability usually becomes more stable and approaches a fixed value.
- Probability can be expressed as a fraction, a decimal or a percentage.
Formulas and facts to remember
Experimental probability of an event E:
P(E) = (Number of trials in which E happened) / (Total number of trials)
This tells you the fraction of times the event occurred out of all the trials you performed.
Range of probability:
0 ≤ P(E) ≤ 1
A probability of 0 means the event never happened in any trial; a probability of 1 means it happened in every trial.
Sum rule (for complementary events):
P(E) + P(not E) = 1
If you know how often something happened, subtract from 1 to find how often it did not happen.
Converting forms:
To convert a fraction to a percentage, multiply by 100. For example, 3/10 = 0.3 = 30 %.
Worked examples
### Example 1: Tossing a coin
Rohan tosses a one-rupee coin 50 times and records 28 heads and 22 tails. Find the experimental probability of getting a head.
Solution:
Total trials = 50
Number of heads = 28
Experimental probability of head = 28/50 = 14/25 = 0.56
So the probability of getting a head, based on this experiment, is 14/25 or 0.56.
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### Example 2: Drawing coloured marbles
A bag contains some red, blue and green marbles. Sneha draws one marble, notes its colour, returns it, and repeats this 80 times. She gets red 24 times, blue 36 times and green 20 times. What is the experimental probability of drawing (i) a blue marble, (ii) a marble that is not green?
Solution:
Total trials = 80
(i) Blue appeared 36 times.
P(blue) = 36/80 = 9/20 = 0.45
(ii) Marbles that are not green = red + blue = 24 + 36 = 60
P(not green) = 60/80 = 3/4 = 0.75
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### Example 3: Quality check in a factory
A small bulb factory tests 200 bulbs and finds that 14 are defective. Estimate the probability that a bulb chosen at random from production is defective.
Solution:
Total bulbs tested = 200
Defective bulbs = 14
P(defective) = 14/200 = 7/100 = 0.07
So the experimental probability that a randomly chosen bulb is defective is 7/100, or 7 %.
Common mistakes
- Counting only favourable trials and forgetting to divide by total trials → Always use the formula: favourable trials divided by total trials.
- Writing probability greater than 1, such as 120/100 → Check that your numerator never exceeds your denominator; probability cannot exceed 1.
- Confusing outcomes with events → An outcome is a single result; an event may include several outcomes (e.g., getting an even number on a die is the event containing 2, 4, 6).
- Assuming experimental probability is exact → Remember it is an estimate based on trials; more trials generally give a better estimate.
- Mixing up "not E" calculations → Subtract the probability of E from 1; do not subtract the frequency from 1.
Quick revision
- Experimental probability = (favourable trials) / (total trials).
- Probability always lies between 0 and 1.
- P(E) + P(not E) = 1.
- More trials usually lead to a more stable probability estimate.
- Express probability as a fraction, decimal or percentage — all are correct.
- A random experiment is one whose result cannot be known in advance.