CIE IGCSE Mathematics (0580) Introduction to probability Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Introduction to probability Study Notes
LEARNING OBJECTIVE
- Understanding the Probability Scale (0 to 1)
Key Concepts:
- Understanding the Probability Scale (0 to 1)
Understanding the Probability Scale (0 to 1)
Understanding the Probability Scale (0 to 1)
Probability is a measure of how likely an event is to occur. It is always a number between 0 and 1.
- A probability of 0 means the event is impossible.
- A probability of 1 means the event is certain.
- A probability of 0.5 means the event is equally likely to happen or not happen.
Probabilities can also be expressed as fractions or percentages.
Example :
A coin is tossed. What is the probability of getting a head?
▶️ Answer/Explanation
There are two equally likely outcomes: Head or Tail.
\( \text{Probability of Head} = \dfrac{1}{2} = 0.5 \)
This lies exactly in the middle of the probability scale.
Example :
What is the probability of rolling a 7 on a standard six-sided die?
▶️ Answer/Explanation
The possible outcomes are: 1, 2, 3, 4, 5, 6.
Since 7 is not a possible outcome, the probability is:
\( \text{Probability} = 0 \)
This is an impossible event.
Example :
A bag contains only red balls. What is the probability of picking a red ball?
▶️ Answer/Explanation
Since all balls are red, picking a red ball is certain.
\( \text{Probability} = 1 \)
Understanding and Using Probability Notation
Understanding and Using Probability Notation
In probability, we use standard notation to represent the likelihood of events. The notation makes it easier to write and solve probability problems.
Probability Notation:
| Symbol | Meaning |
|---|---|
| \( P(A) \) | Probability of event A occurring |
| \( P(\text{not A}) \) | Probability that event A does not occur |
| \( P(A’) \) | Another notation for “not A” (complement of A) |
| \( 0 \leq P(A) \leq 1 \) | Probability is always between 0 and 1 |
Example :
A die is rolled. Let A be the event “rolling an even number”. Write \( P(A) \).
▶️ Answer/Explanation
Even numbers on a die: 2, 4, 6 → 3 outcomes
Total possible outcomes: 6
\( P(A) = \dfrac{3}{6} = 0.5 \)
Example :
In a bag of 5 red and 3 blue balls, let B be the event “picking a blue ball”. Find \( P(B) \) and \( P(\text{not B}) \).
▶️ Answer/Explanation
Total balls = 5 + 3 = 8
\( P(B) = \dfrac{3}{8} \), since there are 3 blue balls
\( P(\text{not B}) = 1 – \dfrac{3}{8} = \dfrac{5}{8} \), which is the probability of getting a red ball
Calculating the Probability of a Single Event
Calculating the Probability of a Single Event
To calculate the probability of a single event, use the formula:
\( P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}} \)
The answer can be written as a fraction, decimal, or percentage.
Example :
A bag contains 4 red balls, 3 blue balls and 5 green balls. What is the probability of picking a red ball?
▶️ Answer/Explanation
Total balls = 4 + 3 + 5 = 12
Favourable outcomes (red balls) = 4
\( P(\text{red}) = \dfrac{4}{12} = \dfrac{1}{3} = 0.333… \approx 33.3\% \)
Example :
The table shows the number of pets owned by 30 students:
| Pets | Number of Students |
|---|---|
| Dog | 12 |
| Cat | 8 |
| None | 10 |
What is the probability that a student chosen at random owns a cat?
▶️ Answer/Explanation
Total students = 30
Favourable outcomes = 8
\( P(\text{cat}) = \dfrac{8}{30} = \dfrac{4}{15} \approx 0.267 \approx 26.7\% \)
Example:
A class of 40 students were surveyed. The Venn diagram shows the number of students who like tea (T) and coffee (C)
What is the probability that a student chosen at random likes tea?
▶️ Answer/Explanation
Let’s say from the diagram:
- 10 like only tea
- 8 like both tea and coffee
- Total who like tea = 10 + 8 = 18
\( P(\text{tea}) = \dfrac{18}{40} = \dfrac{9}{20} = 0.45 = 45\% \)
Probability of an Event Not Occurring
Probability of an Event Not Occurring
The probability of an event not happening is calculated by subtracting the probability of it happening from 1.
$ P(\text{not A}) = 1 – P(A)$
This is useful when it’s easier to find the probability of an event happening and subtract from 1.
Example:
A die is rolled. What is the probability of not rolling a 6?
▶️ Answer/Explanation
\( P(\text{rolling a 6}) = \dfrac{1}{6} \)
\( P(\text{not rolling a 6}) = 1 – \dfrac{1}{6} = \dfrac{5}{6} \)
Example:
The probability that a student walks to school is 0.7. What is the probability that the student does not walk to school?
▶️ Answer/Explanation
\( P(\text{not walking}) = 1 – 0.7 = 0.3 \)
Example:
In a survey of 80 students, 56 said they preferred science over humanities. What is the probability that a randomly chosen student does not prefer science?
▶️ Answer/Explanation
\( P(\text{prefers science}) = \dfrac{56}{80} = 0.7 \)
\( P(\text{does not prefer science}) = 1 – 0.7 = 0.3 = \dfrac{24}{80} = 30\% \)