Chance and Probability ICSE Class 8: Coins, Dice and the 0 to 1 Scale

What Chance and Probability is about

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Chance and Probability sits in the Data Handling theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Random experiments and outcomes

A random experiment has several possible results and you cannot say in advance which one will happen.

List all outcomes first (the sample space):

• tossing a coin: $\{H,T\}$, 2 outcomes;
• throwing a die: $\{1,2,3,4,5,6\}$, 6 outcomes;
• tossing two coins: $\{HH,HT,TH,TT\}$, 4 outcomes (HT and TH are different);
• picking a ball from a bag: one outcome for each ball.

If the coin or die is fair (unbiased), all its outcomes are equally likely.

Worked example

List the outcomes when two coins are tossed together. How many outcomes have exactly one head?

Try it first, then show the solutionSolution

1. First coin H or T, second coin H or T: $HH,HT,TH,TT$.
2. Exactly one head: $HT$ and $TH$, so 2 outcomes.

Answer: 4 outcomes; 2 with exactly one head

Finding a probability

For equally likely outcomes:
$P(E)=\frac{\text{favourable outcomes}}{\text{total outcomes}}$.

Steps:
1. Count the total number of outcomes.
2. List and count the outcomes that give the event.
3. Divide, and simplify the fraction.

Probability can be written as a fraction, a decimal or a percentage: $\frac{1}{2}=0.5=50\%$.

Worked example

A fair die is thrown. Find the probability of getting an even number.

Try it first, then show the solutionSolution

1. Total outcomes: 6.
2. Even numbers: 2, 4, 6, which is 3 outcomes.
3. $P=\frac{3}{6}=\frac{1}{2}$.

Answer: $\frac{1}{2}$

The probability scale from 0 to 1

Since favourable outcomes are never fewer than 0 or more than the total, every probability lies between 0 and 1: $0\le P(E)\le 1$.

Probability	Meaning	Example (one die)
0	impossible	getting a 7
near 0	unlikely	getting a 6 ($\frac{1}{6}$)
$\frac{1}{2}$	as likely as not	getting an odd number
near 1	likely	getting less than 6 ($\frac{5}{6}$)
1	certain	getting less than 7

Worked example

On a die, find the probability of getting (i) a 7 (ii) a number less than 7. Describe each event.

Try it first, then show the solutionSolution

1. (i) No outcome is 7: $P=\frac{0}{6}=0$, impossible.
2. (ii) All 6 outcomes are less than 7: $P=\frac{6}{6}=1$, certain.

Answer: (i) 0, impossible (ii) 1, certain

Probability of an event not happening

An event either happens or it does not, so
$P(\text{not }E)=1-P(E)$.

This is quicker when the 'not' event has many outcomes. For example, the probability of not getting a 6 on a die is $1-\frac{1}{6}=\frac{5}{6}$.

Also, the probabilities of all the separate outcomes of an experiment add up to 1.

Worked example

The probability that a school bus is late on a given day is 0.42. What is the probability that it is not late?

Try it first, then show the solutionSolution

1. $P(\text{not late})=1-0.42$.
2. $=0.58$.

Answer: 0.58

Bags, cards and spinners

Picking one item at random means every item is equally likely.

• Bag of balls: total = all the balls; favourable = balls of the colour asked for.
• Numbered cards: list the numbers that fit (even, prime, multiple of 5, perfect square).
• Spinner with equal sectors: each sector is one outcome.

Remember: 1 is not a prime number.

Worked example

A bag has 6 red, 4 blue and 2 green balls. One is drawn at random. Find P(red), P(blue) and P(not green).

Try it first, then show the solutionSolution

1. Total $=12$.
2. $P(\text{red})=\frac{6}{12}=\frac{1}{2}$; $P(\text{blue})=\frac{4}{12}=\frac{1}{3}$.
3. $P(\text{not green})=\frac{10}{12}=\frac{5}{6}$ (or $1-\frac{2}{12}$).

Answer: $\frac{1}{2}$, $\frac{1}{3}$, $\frac{5}{6}$

Experimental probability and chance in life

If you actually do an experiment many times:
$\text{experimental probability}=\frac{\text{number of times the event happened}}{\text{number of trials}}$.

The experimental value is usually close to, but not exactly, the theoretical value. With more trials it tends to get closer.

In daily life, probability describes things like a weather forecast ("70% chance of rain") or the chance of winning a lucky draw. A 70% chance of rain means rain is likely, not certain.

Worked example

A coin is tossed 50 times and shows heads 28 times. Find the experimental probability of heads and compare it with the theoretical value.

Try it first, then show the solutionSolution

1. Experimental $=\frac{28}{50}=\frac{14}{25}=0.56$.
2. Theoretical $=\frac{1}{2}=0.5$; the experimental value is close but not equal.

Answer: 0.56, close to 0.5

Maths in real life: Chance and Probability

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.

Topics in this chapter

• Probability of an Event: Coins, Dice and Bags
• Probability Scale and Experimental Probability

Chance and Probability formula sheet

Probability

$P(E)=\frac{\text{favourable}}{\text{total}}$

Equally likely outcomes.

Range

$0\le P(E)\le 1$

0 impossible, 1 certain.

Not happening

$P(\text{not }E)=1-P(E)$

Complement.

Experimental

$\frac{\text{times event occurred}}{\text{number of trials}}$

From real trials.

How the school exam asks Chance and Probability

The question patterns to expect, and the method that works for each.

Tricks to remember Chance and Probability

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

• Want over all

  Probability = outcomes you want ÷ all outcomes.

  Use it when: Every calculation.

• Zero to one, never more

  If you get a probability above 1 or below 0, something is wrong.

  Use it when: Checking answers.

• Not = 1 minus

  Chance of 'not' is one minus the chance of 'yes'.

  Use it when: Complement questions.

• 1 is not prime

  Primes on a die are 2, 3 and 5 only.

  Use it when: Prime-number events.

Chance and Probability: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Chance and Probability

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

• ✗Mistake: Counting 1 as prime

  A prime has exactly two factors; 1 has only one.

  Wrong working

  Primes on a die: 1, 2, 3, 5

  Right working

  2, 3, 5

  ✓Do this: Start primes at 2.

  Usually costs: The answer.

• ✗Mistake: Treating HT and TH as the same

  With two coins, 'one head' can happen in two ways.

  Wrong working

  Outcomes: two heads, one head, no head, so P(one head) $=\frac{1}{3}$

  Right working

  $HH,HT,TH,TT$: $P(\text{one head})=\frac{1}{2}$

  ✓Do this: List outcomes for each coin separately.

  Usually costs: Most of the marks.

• ✗Mistake: A probability greater than 1

  Usually the total and favourable counts are swapped.

  ✓Do this: Favourable goes on top; check the answer is between 0 and 1.

  Usually costs: The answer.

• ✗Mistake: Not simplifying

  $\frac{3}{6}$ should be given as $\frac{1}{2}$.

  ✓Do this: Reduce fractions to lowest terms.

  Usually costs: Presentation mark.

• ✗Mistake: Mixing experimental and theoretical

  Results of actual trials are experimental; they need not equal the theoretical value.

  ✓Do this: Label which probability you are finding.

  Usually costs: 1 mark.

Practice questions on Chance and Probability

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

1. Q1 · 1 mark · MCQExam level

   The probability of an impossible event is:
   (a) 0
   (b) $\frac{1}{2}$
   (c) 1
   (d) $-1$

   Hint

   It never happens.

   Answer

   (a)

2. Q2 · 1 mark · Very short answerExam level

   A fair coin is tossed. What is the probability of getting a tail?

   Hint

   Two outcomes.

   Answer

   $\frac{1}{2}$.

3. Q3 · 2 marks · Short answerExam level

   A die is thrown. Find the probability of getting a number less than 3.

   Hint

   1 and 2.

   Answer

   $\frac{2}{6}=\frac{1}{3}$.

4. Q4 · 3 marks · Short answerHarder exam level

   A letter is chosen at random from the word PROBABILITY. Find the probability that it is (i) B (ii) a vowel.

   Hint

   11 letters; vowels O, A, I, I.

   Answer

   (i) $\frac{2}{11}$ (ii) $\frac{4}{11}$.

5. Q5 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): The probability of an event cannot be greater than 1.
   Reason (R): The number of favourable outcomes can never exceed the total number of outcomes.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Look at the fraction.

   Answer

   (c)

6. Q6 · 2 marks · Short answerHarder exam level

   A spinner has 8 equal sectors numbered 1 to 8. Find the probability that it stops on (i) an odd number (ii) a number greater than 6.

   Hint

   8 outcomes.

   Answer

   (i) $\frac{4}{8}=\frac{1}{2}$ (ii) $\frac{2}{8}=\frac{1}{4}$.

7. Q7 · 1 mark · Very short answerHarder exam level

   If the probability of an event is 0.35, find the probability that it does not happen.

   Hint

   1 − 0.35.

   Answer

   0.65.

8. Q8 · 2 marks · Short answerHarder exam level

   A box has 12 bulbs, of which 3 are faulty. One bulb is picked at random. Find the probability that it is good.

   Hint

   9 good bulbs.

   Answer

   $\frac{9}{12}=\frac{3}{4}$.

9. Q9 · 4 marks · Long answerHarder exam level

   Two dice are thrown together (36 outcomes). Find the probability of (i) a total of 7 (ii) a double (both dice the same).

   Hint

   List pairs.

   Answer

   (i) (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): $\frac{6}{36}=\frac{1}{6}$. (ii) (1,1) to (6,6): $\frac{6}{36}=\frac{1}{6}$.

10. Q10 · 2 marks · Short answerHarder exam level

    A coin is tossed 200 times and shows heads 108 times. Find the experimental probability of (i) heads (ii) tails.

    Hint

    Tails = 200 − 108.

    Answer

    (i) $\frac{108}{200}=0.54$ (ii) $\frac{92}{200}=0.46$.

11. Q11 · 3 marks · Short answerHOTS / competency

    A bag has 5 red balls and some blue balls. The probability of drawing a blue ball is $\frac{2}{3}$. How many blue balls are there?

    Hint

    P(red) = ⅓.

    Answer

    $P(\text{red})=\frac{1}{3}$, so the total is 15 and blue $=10$.

12. Q12 · 3 marks · Short answerHOTS / competency

    A number is chosen at random from 1 to 50. Find the probability that it is a perfect square.

    Hint

    1, 4, 9, …

    Answer

    1, 4, 9, 16, 25, 36, 49: $\frac{7}{50}$.

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. 1.What is the probability of an impossible event?Show answer

   0.

2. 2.What is the probability of a certain event?Show answer

   1.

3. 3.P(head) for a fair coin?Show answer

   ½.

4. 4.P(4) on a fair die?Show answer

   ⅙.

5. 5.If P(E) = 0.3, what is P(not E)?Show answer

   0.7.

6. 6.How many outcomes when two coins are tossed?Show answer

   4.

7. 7.Probability near 0 means the event is…?Show answer

   Unlikely.

8. 8.Primes from 1 to 10?Show answer

   2, 3, 5, 7.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Did you list all the outcomes first?
• ☐Did you count HT and TH separately?
• ☐Is your answer between 0 and 1?
• ☐Did you leave 1 out of the primes?
• ☐Did you use 1 − P(E) where it is quicker?
• ☐Is your fraction in lowest terms?

Chance and Probability: frequently asked questions