1. Meaning of Probability

2. The Meaning of Probability
You take turn to throw the dice and see who gets a larger number.
Let’s play a game using this dice:
I don’t know whether I can get a number greater than 4.
I get a ‘4’.
Similar to this case, we often come across situations which involve uncertain results.

3. Let’s study the following examples.
For activities with uncertain results, a particular result that we are interested in is called an event.
Let’s study the following examples.
This event will never happen.
This event is very likely to happen.
This event will certainly happen.
Activity: Draw a marble from a box with 99 red marbles and 1 black marble.
Activity: Draw a card from 52 playing cards.
Activity: Throw a dice.
Event: A number less than ‘7’ is obtained.
Event: A green heart is obtained.
Event: A red marble is drawn.

4. Activity: Throw a dice.
Activity: Draw a marble from the box below.
Activity: Draw a card from 52 playing cards.
Event: A number less than ‘7’ is obtained.
Event: A red marble is drawn.
Event: A green heart is obtained.
certainly happen
very likely to happen
never happen
In mathematics, the chance of occurrence of an event is represented by a number which is known as the probability of the event.
The higher the chance of occurrence, the greater is the probability.

5. Let’s consider one of the previous activities.
The chance of drawing a red marble is higher than that of drawing a black marble.
That means, the probability of drawing a red marble is greater than that of drawing a black marble.
Let’s consider one of the previous activities.
Activity: Draw a marble from a box with 99 red marbles and 1 black marble.
Event: A red marble is drawn.

6. Possible Outcomes What are the possible results of tossing a coin?
You are right ! A ‘Head’ and a ‘Tail’ are the two possible outcomes.
A coin has 2 faces, a ‘Head’ and a ‘Tail’.
Thus, the possible results of tossing a coin are a ‘Head’ and a ‘Tail’.
a ‘Head’
a ‘Tail’

7. The collection of all possible outcomes is called the sample space.
So, a ‘Head’ and a ‘Tail’ make up the sample space of tossing a coin.

8. For tossing a fair coin, the chance of obtaining a ‘Head’ and a ‘Tail’ are equal, so the two outcomes are called equally likely outcomes.
For throwing a fair dice, sample space: _____________________ They ( are / are not ) equally likely outcomes.
‘1’, ‘2’, ‘3’, ‘4’, ‘5’ and ‘6’

9. Calculation of Probabilities
Activity: Throw a dice
Event: Get a number less than 3
‘1’ and ‘2’ are the two favourable outcomes.
Among all possible outcomes, those favourable to the event are called favourable outcomes.
List out all the possible outcomes and the favourable outcomes of the following.
Activity & Event
Possible outcomes
Favourable outcomes
Randomly choose a card from a pack of 52 playing cards & obtain a ‘Q’
Q, Q, Q and Q
All of the 52 cards

10. The following shows how to calculate the probability of an event.
The probability of an event E, denoted by P(E), is given by
Note that in this definition, all the possible outcomes are equally likely.

11. Consider the event of throwing a fair dice and obtaining a number less than 3.
No. of all possible outcomes = 6
No. of favourable outcomes = 2 2 6
∴ P(obtaining a number less than 3)
i.e. the probability of getting a number less than 3 is .

12. Follow-up question
A letter is chosen at random from the word ‘PROBABILITY’. Find the probabilities of getting the letter (a) ‘A’, (b) ‘B’.
Solution
(a)
∵ There is only 1 ‘A’ among the 11 letters. ∴
(b)
∵ There are 2 ‘B’s among the 11 letters. ∴

13. Impossible Events and Certain Events
In probability, an event which is impossible to occur is called an impossible event.
Example: Throw a dice.
Since all numbers on a dice are not greater than 6,
it is impossible to get a number greater than 6.
 There are 0 favourable outcomes of getting a number greater than 6. ∴

14. An event which is certain to occur is called a certain event.
Example: Throw a dice.
Since all numbers on a dice are less than 7,
the number we get is certainly less than 7. ∴

15. From the above, we can see that
P(impossible event) = 0
P(certain event) = 1
For any event E, the probability of its occurrence P(E) must satisfy: