MathsICSE Class 8 · Chance and Probability

The Probability Scale and Experimental Probability (ICSE Class 8 Maths)

Q: Is experimental probability the same as theoretical?

Not exactly; they get closer with more trials.

Updated 17 October 2026 · Fact-checked

Probability lies on a scale from $0$ (impossible) to $1$ (certain); near $0$ is unlikely, near $\frac{1}{2}$ is equally likely, near $1$ is likely. Experimental probability is $\frac{\text{number of times the event occurred}}{\text{total number of trials}}$, and the expected number of occurrences in $N$ trials is $P \times N$.

Understand Probability Scale and Experimental Probability

The CISCE curriculum links chance to life events: probability near $0$ is unlikely, around $\frac{1}{2}$ is equally likely, and near $1$ is likely.

The scale. Mark $0$ for "impossible", $\frac{1}{2}$ for "even chance" and $1$ for "certain". Probabilities between are words like unlikely ( $0.1$ ), likely ( $0.9$ ). For example, $0.97$ means very likely.

Experimental (empirical) probability. Instead of counting equally likely outcomes, repeat the experiment many times and record:
$ $P (E) \approx \frac{times E occurred}{total trials} .$ $
The more trials, the closer it gets to the theoretical value (for a fair coin it tends to $\frac{1}{2}$ ).

Expected number. If $P (E)$ is the chance, then in $N$ trials about $P (E) \times N$ occurrences are expected. This is a prediction, not a guarantee.

Complement. $P (E) + P (not E) = 1$ applies to experimental probabilities too.

Frequency tables. For a die rolled 60 times, the frequencies add to $60$ ; the experimental probability of each face is its frequency over $60$ , and the theoretical value is $\frac{1}{6}$ .

Changing contents. Adding or removing items changes both the favourable count and the total, so recompute the fraction. In a bag with $3$ red and $5$ white balls, adding $3$ white balls changes $P (white)$ from $\frac{5}{8}$ to $\frac{8}{11}$ .

Probability Scale and Experimental Probability: formulas to remember

Experimental probability: $P (E) = \frac{times E occurred}{total trials}$; Also called relative frequency.
Expected number: $expected = P (E) \times N$; For $N$ trials.
Complement: $P (E) + P (not E) = 1$; Holds for any probabilities.

How to solve Probability Scale and Experimental Probability questions

Record trials in a table, form the fraction, and use it to predict.

1Record the number of trials and the number of times the event occurred.
2Write $\frac{occurred}{trials}$ in lowest terms.
3Use the complement for "does not happen".
4Multiply by $N$ for expected counts.
5Compare with the theoretical value if it is known.

Probability Scale and Experimental Probability in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Defective items

Suppose in a sample of 200 packets from a factory, 14 are found defective.

Estimate the number of defective packets in a batch of 1500.

Try it first, then show the solutionSolution

$P (defective) = \frac{14}{200} = \frac{7}{100} = 0.07$ .
Expected defective $= 0.07 \times 1500 = 105$ .

Answer: About 105 packets.

Memory hook: Sample fraction times the batch size.

Tricks to remember Probability Scale and Experimental Probability

Short memory aids. Say each one aloud once, then use it on a question the same day.

Near 0, near half, near 1
Think of the scale: near 0 unlikely, 0.5 equally likely, near 1 likely.
Use it when: Verbal questions.
Trials in the denominator
Experimental probability always has the number of trials, not the number of outcomes, below.
Use it when: Frequency table questions.

Probability Scale and Experimental Probability: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Reading the scale · 1 mark · MCQBoard level

A probability of 0.97 means the event is:
(a) impossible (b) unlikely (c) equally likely (d) very likely

Try it first, then show the solutionSolution

$0.97$ is close to $1$ .

Answer: (d)

Example 2: Coin trials · 3 marks · Short answerBoard level

A coin is tossed 50 times and a head appears 29 times. Find the experimental probability of a head and of a tail.

Try it first, then show the solutionSolution

$P (head) = \frac{29}{50} = 0.58$ .
Tails $= 50 - 29 = 21$ ; $P (tail) = \frac{21}{50} = 0.42$ .
Check: $0.58 + 0.42 = 1$ .

Answer: 0.58; 0.42

Example 3: Die trials · 3 marks · Short answerHarder board level

A die is rolled 60 times with results: 1 (8 times), 2 (11), 3 (9), 4 (12), 5 (10), 6 (10). Find the experimental probability of (i) a six, (ii) an even number, and compare with the theoretical values.

Try it first, then show the solutionSolution

Check: $8 + 11 + 9 + 12 + 10 + 10 = 60$ .
(i) $\frac{10}{60} = \frac{1}{6}$ (equal to the theoretical $\frac{1}{6}$ ).
(ii) Even: $11 + 12 + 10 = 33$ ; $\frac{33}{60} = \frac{11}{20} = 0.55$ against the theoretical $\frac{1}{2}$ .

Answer: $\frac{1}{6}$ ; $\frac{11}{20}$

Example 4: Bag counts from probabilities · 4 marks · Short answerHarder board level

In a bag, $P (red) = 0.4$ and $P (blue) = 0.25$ , the rest being green. There are 80 balls. Find the number of balls of each colour and $P (green)$ .

Try it first, then show the solutionSolution

$P (green) = 1 - 0.4 - 0.25 = 0.35$ .
Red $= 0.4 \times 80 = 32$ ; blue $= 0.25 \times 80 = 20$ ; green $= 0.35 \times 80 = 28$ .
Check: $32 + 20 + 28 = 80$ .

Answer: Red 32, blue 20, green 28; $P (green) = 0.35$

Example 5: Changing the bag · 4 marks · Long answerHOTS / competency

A bag has 3 red and 5 white balls. How many white balls must be added so that $P (white) = \frac{4}{5}$ ? Verify.

Try it first, then show the solutionSolution

Add $w$ white balls: $\frac{5 + w}{8 + w} = \frac{4}{5}$ .
$25 + 5 w = 32 + 4 w \Rightarrow w = 7$ .
Check: $\frac{12}{15} = \frac{4}{5}$ .

Answer: Add 7 white balls

Where students lose marks in Probability Scale and Experimental Probability

The slips examiners see most often, and what to write instead.

Mistake: Using outcomes instead of trials in the denominator
Experimental probability divides by the number of trials.
Wrong working
29 heads in 50 tosses: $\frac{29}{2}$ .
Right working
$\frac{29}{50}$ .
Do this: Write "trials" below the fraction.
Usually costs: Answer mark.
Mistake: Treating an expected count as certain
It is a prediction; the actual count may differ.
Wrong working
"There will be exactly 105 defective packets."
Right working
"About 105 are expected."
Do this: Use the word "expected".
Usually costs: Interpretation mark.

Exam tips

Always check that frequencies add up to the number of trials.
Write whether an event is likely or unlikely using the scale.
For changing contents, recompute both numerator and denominator.

Practice questions on Probability Scale and Experimental Probability

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If the chance of rain tomorrow is 0.35, the chance that it does not rain is:
(a) 0.35 (b) 0.65 (c) 1.35 (d) 0.5
Hint
Complement.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
$P (E) = \frac{7}{20}$ . Find $P (not E)$ and say whether $E$ is likely or unlikely.
Hint
Subtract from 1.
Answer
$P (not E) = \frac{13}{20}$ . Since $\frac{7}{20} = 0.35 < 0.5$ , $E$ is unlikely.
Q3 · 3 marks · Short answerHarder board level
Of 200 packets, 14 are defective. Estimate the number of defective packets among 1500, and the experimental probability that a packet is not defective.
Hint
Probability from the sample.
Answer
$P (defective) = 0.07$ , expected $= 105$ . $P (not defective) = 0.93$ .
Q4 · 3 marks · Short answerHarder board level
A spinner is spun 40 times and lands on blue 12 times. Find the experimental probability of blue, and the expected number of blues in 500 spins.
Hint
12 over 40.
Answer
$P = \frac{12}{40} = \frac{3}{10}$ . Expected $= \frac{3}{10} \times 500 = 150$ .
Q5 · 4 marks · Long answerHOTS / competency
A box has 12 cards. $P (red) = \frac{1}{4}$ and $P (blue) = \frac{1}{3}$ ; the rest are green. Find the number of cards of each colour, and $P (green)$ after 3 green cards are removed.
Hint
Multiply each probability by 12.
Answer
Red $= 3$ , blue $= 4$ , green $= 12 - 7 = 5$ . After removing $3$ green cards: green $= 2$ out of $9$ , so $P (green) = \frac{2}{9}$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Probability scale endpoints?Show answer
0 impossible and 1 certain.
2.Experimental probability formula?Show answer
Times the event occurred over total trials.
3.Expected number in N trials?Show answer
P(E) x N.
4.Does more trials help?Show answer
Yes, the experimental value gets closer to the theoretical one.

The Probability Scale and Experimental Probability (ICSE Class 8 Maths)

Understand Probability Scale and Experimental Probability

Probability Scale and Experimental Probability: formulas to remember

How to solve Probability Scale and Experimental Probability questions

Probability Scale and Experimental Probability in real life

Tricks to remember Probability Scale and Experimental Probability

Probability Scale and Experimental Probability: solved examples

Where students lose marks in Probability Scale and Experimental Probability

Exam tips

Practice questions on Probability Scale and Experimental Probability

Test your recall

Probability Scale and Experimental Probability: frequently asked questions

Is experimental probability the same as theoretical?

Can probability be above 1?