Meaning of Probability

Probability is the branch of mathematics that deals with the study of chance and uncertainty.
It tells us how likely an event is to occur.

We use probability to describe situations where we are not certain about the outcome — for example:

1. Will it rain tomorrow?


2. Will our school team win the next match?


3. If I pick a card, what are the chances it will be red?

In short, probability measures how likely or unlikely something is to happen.

1. Predicting if it will rain or not.


2. Tossing a coin and expecting heads or tails.


3. Drawing a card from a pack.


4. Selecting a student at random to answer a question.


5. Checking the chance that a bus will arrive late.


6. Expecting your name to be chosen in a school raffle draw.


7. Guessing whether a baby will be a boy or girl.


8. Choosing a colour of ball from a bag without looking.


9. Estimating the chance of passing an exam based on past results.


10. Predicting which football team will win a match.

1. Experiment — Any activity or process that can be repeated and has two or more possible outcomes.
   Example: Tossing a coin or drawing a card.


2. Outcome — The result of an experiment.
   Example: Getting “head” when a coin is tossed.


3. Sample Space — The set of all possible outcomes of an experiment.
   Example: When tossing a coin → {Head, Tail}.


4. Event — One or more outcomes of interest from an experiment.
   Example: Getting an even number when a die is rolled.


5. Favourable Outcomes — The outcomes we are looking for.
   Example: Getting a 4 when rolling a die — only one favourable outcome.


6. Trial — Each performance or repetition of an experiment.
   Example: Tossing a coin once is one trial.


7. Impossible Event — An event that cannot happen.
   Example: Getting a number greater than 6 on a die.


8. Certain Event — An event that must happen.
   Example: Getting a number less than 7 when rolling a die (since all numbers 1–6 are less than 7).


9. Equally Likely Events — Events that have the same chance of occurring.
   Example: When tossing a fair coin, getting heads or tails are equally likely.

Formula for Probability

The probability of an event is given by:

P(E) = Number of favourable outcomes / Total number of possible outcomes

Where:

1. P(E) = probability of the event E occurring.


2. The result is always between 0 and 1.

Probability Scale

Event Type	Meaning	Probability
Impossible	Cannot happen	0
Unlikely	May not happen	Between 0 and 0.5
Even Chance	Can happen or not happen	0.5
Likely	Expected to happen	Between 0.5 and 1
Certain	Must happen	1

0 means the event will never happen.
1 means the event will definitely happen.
Values between 0 and 1 show how likely it is.

1. Theoretical Probability
   
   – Found using reasoning or calculation, without actual experiment.
   
   – Based on what should happen in theory.
   
   – Example: The probability of getting a head when tossing a fair coin is ½.


2. Experimental Probability
   
   – Found by carrying out an actual experiment or observation.
   
   – Based on what really happens when the experiment is performed.
   
   – Formula:
   Experimental Probability = Number of times event occurs / Total number of trials

1. The probability of any event lies between 0 and 1. (That is, 0 ≤ P(E) ≤ 1).


2. The sum of probabilities of all possible outcomes is 1.


3. The probability of an event not happening is given by: P(not E) = 1 − P(E)


4. If an event is certain, P(E) = 1.


5. If an event is impossible, P(E) = 0.

1. It helps us to make better predictions and decisions.


2. It helps in planning — like estimating risks and chances of success.


3. It helps in games and sports for predicting possible outcomes.


4. It is used in business and insurance to calculate risks and profits.


5. It helps in weather forecasting.


6. It is used in medicine for predicting success of treatment.


7. It helps in education — teachers can estimate the chance of students passing exams.


8. It helps in transportation planning — e.g., predicting bus delays.

1. Makes decision-making easier and more logical.


2. Helps to estimate possible risks and benefits.


3. Useful in everyday life situations.


4. Helps in reasoning and problem-solving.

1. Results are not always certain, only likely.


2. It depends on the accuracy of information or experiment.


3. Incomplete or biased data can lead to wrong results.


4. Some real-life events are too complex to calculate exactly.

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Practice Questions

Probability = (Number of favourable outcomes) ÷ (Total number of possible outcomes)

1) Rainy Days (Weather)

Problem: Over the last 10 days it rained on 7 days. If you pick one of those days at random, what is the probability it was a rainy day?

Steps:
Total days = 10.
Favourable (rain) = 7.
Probability = 7 ÷ 10 = 7/10.
As a percent = 70%.
Answer: 7/10 (70%).

2) Exam Pass (Class)

Problem: In a class of 40 students, 28 passed English. If the teacher picks one student at random, what is the probability the student passed English?

Steps:
Total students = 40.
Favourable (passed English) = 28.
Probability = 28 ÷ 40 = divide by 4 → 7 ÷ 10 = 7/10.
Percent = 70%.
Answer: 7/10 (70%).

3) School Football Team (Sports)

Problem: The school team played 12 matches this season: they won 7, drew 3, lost 2. What is the probability that a randomly chosen match was a win?

Steps:
Total matches = 12.
Favourable (win) = 7.
Probability = 7 ÷ 12 → 7/12 ≈ 58.3%.
Answer: 7/12 (≈58.3%).

4) Market Tomatoes (Food)

Problem: A trader has 50 tomatoes; 15 are rotten and 35 are good. If you pick one tomato without looking, what is the probability it is good?

Steps:
Total tomatoes = 50.
Favourable (good) = 35.
Probability = 35 ÷ 50 = divide by 5 → 7 ÷ 10 = 7/10.
Answer: 7/10 (70%).

5) Bus Arrival (Transport)

Problem: A commuter notes that in 20 days the local bus arrived late on 5 days. If we choose one of those 20 days at random, what is the probability the bus was on time?

Steps:
Total days = 20.
Favourable (on time) = 20 − 5 = 15.
Probability = 15 ÷ 20 = 3 ÷ 4 = 3/4 = 75%.
Answer: 3/4 (75%).

6) Snack Seller (Food Stall)

Problem: A seller has 20 puff-puffs: 8 plain, 7 sugar-coated, 5 banana. You pick one. What is the probability it is sugar-coated?

Steps:
Total = 20.
Favourable (sugar-coated) = 7.
Probability = 7 ÷ 20 = 7/20 = 35%.
Answer: 7/20 (35%).

7) Library Book (School)

Problem: The school library shelf has 3 Maths textbooks, 2 English, and 5 story books. A student takes one book at random. What is the probability it is a story book?

Steps:
Total books = 3 + 2 + 5 = 10.
Favourable (story) = 5.
Probability = 5 ÷ 10 = 1/2 = 50%.
Answer: 1/2 (50%).

8) Subject Choice (Students)

Problem: In JSS3, 60 students were asked about their preferred stream: 35 picked Arts, 25 picked Science. What is the probability a student chosen at random picked Science?

Steps:
Total students = 60.
Favourable (Science) = 25.
Probability = 25 ÷ 60 = divide by 5 → 5 ÷ 12 = 5/12 ≈ 41.7%.
Answer: 5/12 (≈41.7%).

9) School Raffle (Lottery)

Problem: A school sells 100 raffle tickets and will draw 3 winning tickets. If you buy one ticket, what is the probability you win at least one prize?

Steps:
Total tickets = 100. Number of winners = 3.
Probability = 3 ÷ 100 = 3/100 = 0.03 = 3%.
Answer: 3/100 (3%).

10) Fruit Box — Two without Replacement (Shopping)

Problem: A box has 3 mangoes, 4 oranges, 3 apples (total 10 fruits). You pick 2 fruits one after the other without replacement. What is the probability both picked are apples?

Steps:
Total fruits = 10. Favourable first pick (apple) = 3.
P(first apple) = 3 ÷ 10.
After one apple is taken, apples left = 2, total left = 9.
P(second apple | first apple) = 2 ÷ 9.
Multiply: P(both apples) = (3/10) × (2/9) = 6 ÷ 90 = 1 ÷ 15 = 1/15 ≈ 6.67%.
Answer: 1/15 (≈6.67%).

11) School Survey (Union of Events)

Problem: At a club fair, 30 students like football, 20 like basketball, and 10 like both sports. If a student is chosen at random from the group, what is the probability they like football or basketball?

Steps:
P(Football or Basketball) = P(F) + P(B) − P(F and B)
Total unique students = 30 + 20 − 10 = 40.
P(F) = 30 ÷ 40 = 3/4.
P(B) = 20 ÷ 40 = 1/2.
P(F and B) = 10 ÷ 40 = 1/4.
P(F or B) = 3/4 + 1/2 − 1/4 = 1 (100%).
Answer: 1 (100%).

12) School Prize (Conditional Choice)

Problem: In a class of 30, 18 brought books and 12 did not. The teacher picks one student at random who brought books for a prize. What is the probability the picked student brought books?

Steps:
Total students = 30. Favourable (brought books) = 18.
Probability = 18 ÷ 30 = divide by 6 → 3 ÷ 5 = 3/5 = 60%.
Answer: 3/5 (60%).

13) Bus Card Top-up (Percent to Probability)

Problem: A small survey found 60% of commuters use bus card A. If you meet one commuter at random, what is the probability they use bus card A?

Steps:
Percent 60% = 60 ÷ 100 = 0.60.
As fraction = 60/100 = 3/5.
Probability = 3/5 (0.6 or 60%).
Answer: 3/5 (60%).

14) School Canteen (At least one)

Problem: The canteen sells 4 types of drinks; yesterday 40 cartons were sold: 10 cola, 12 malt, 8 juice, 10 water. If you pick one sold carton at random, what is the probability it is not juice?

Steps:
Total = 40. Favourable (not juice) = 40 − 8 = 32.
Probability = 32 ÷ 40 = divide by 8 → 4 ÷ 5 = 4/5 = 80%.
Answer: 4/5 (80%).

15) Two-stage with Replacement (Marbles)

Problem: A jar has 6 red and 4 blue marbles. You draw one marble, note the colour, put it back, then draw again. What is the probability both draws are red?

Steps:
Since we replace, probabilities do not change between draws.
P(red on first) = 6 ÷ 10 = 3/5.
P(red on second) = 3/5.
Multiply: (3/5) × (3/5) = 9/25 = 36%.
Answer: 9/25 (36%).

16) Tossing a Coin

When a coin is tossed:

Sample space = {Head (H), Tail (T)}
Total outcomes = 2
Favourable outcomes for Head = 1

        P(Head) = 1 ÷ 2 = 1/2
        

Similarly,

        P(Tail) = 1 ÷ 2 = 1/2
        

17) Rolling a Die

When a die is rolled:

Sample space = {1, 2, 3, 4, 5, 6}

If we want the probability of getting an even number:

Favourable outcomes = {2, 4, 6} → 3 outcomes

        P(Even number) = 3 ÷ 6 = 1/2
        

18) Picking a Red Ball

A box contains 5 red balls, 3 blue balls, and 2 green balls.
Find the probability of picking:

• A red ball


• A green ball

Solution:

Total number of balls = 5 + 3 + 2 = 10

(i) Favourable outcomes for red = 5

        P(Red) = 5 ÷ 10 = 1/2
        

(ii) Favourable outcomes for green = 2

        P(Green) = 2 ÷ 10 = 1/5
        

19) Getting a 3 on a Die

Sample space = {1, 2, 3, 4, 5, 6}
Favourable outcome = {3}
Total outcomes = 6

        P(3) = 1 ÷ 6
        

Range of Probability

The value of probability is always between 0 and 1.

Probability	Meaning	Example
0	Impossible event	Getting 8 when rolling a die
1/2	Equal chance	Getting Head or Tail
1	Certain event	Getting a number less than 7 on a die

Complementary Events

If the probability of an event happening is P(E), then the probability of it not happening is:

        P(not E) = 1 - P(E)
        

Example: If P(getting a head) = 1/2, then
P(not getting a head) = 1 - 1/2 = 1/2

20) Word Problem — White and Black Balls

A bag contains 4 white balls and 6 black balls. If one ball is picked at random, find:

• (a) The probability of picking a white ball.


• (b) The probability of picking a black ball.

Solution:

Total balls = 4 + 6 = 10

(a) P(White) = 4/10 = 2/5

(b) P(Black) = 6/10 = 3/5

21) Word Problem — Red, Green, Blue Balls

A box contains 2 red, 3 green, and 5 blue balls. If a ball is selected at random, find the probability that it is:

• (a) Red


• (b) Not blue

Solution:

Total = 2 + 3 + 5 = 10

(a) P(Red) = 2/10 = 1/5

(b) Not blue → Red or Green = 2 + 3 = 5
P(Not blue) = 5/10 = 1/2

22) Word Problem — Prime Number or Greater than 4

A die is thrown once. Find the probability of getting:

• (a) A prime number


• (b) A number greater than 4

Solution:

Sample space = {1, 2, 3, 4, 5, 6}

(a) Prime numbers = {2, 3, 5} → 3 outcomes

P(Prime) = 3/6 = 1/2

(b) Numbers greater than 4 = {5, 6} → 2 outcomes

P(Greater than 4) = 2/6 = 1/3

23) Word Problem — Coin Tossed Twice

A coin is tossed twice. Find the probability of getting:

• (a) Two heads


• (b) One head

Solution:

Sample space = {HH, HT, TH, TT} → 4 outcomes

(a) Two heads = {HH} → 1 outcome

P(Two heads) = 1/4

(b) One head = {HT, TH} → 2 outcomes

P(One head) = 2/4 = 1/2

Quick Tips for Solving Real-Life Probability Problems

1. Count the total possible items (people, days, fruits, tickets).


2. Count the favourable cases (the outcomes you want).


3. Divide favourable by total.


4. Simplify the fraction and, if helpful, convert to percent or decimal.


5. Use complement (1 − P(E)) when “not” is easier to count.


6. For sequential draws: if without replacement, reduce the total after each draw; if with replacement, totals stay the same and multiply probabilities.

Summary

1. Probability measures how likely an event is to occur.


2. The value of probability is between 0 and 1.


3. Formula: P(E) = Number of favourable outcomes / Total number of possible outcomes


4. Probability helps us to make informed predictions in uncertain situations.