GCSE Maths

Numbers
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Integers
Integers Quiz
Prime Numbers and Composite Numbers
Positive and Negative Numbers
Order of Operations
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The Laws of Indices
Square Roots and Cube Roots
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
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Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
Expressing Fractions as Decimal Numbers
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Algebra
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Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Substituting Values into Formulae
Rearranging Equations
Algebraic Fractions
Expanding Brackets and Simplifying
Expanding Double Brackets
Factorisation
Factorising Quadratic Expressions
Factorising Quadratics with a Coefficient of x Squared Greater than 1
The Quadratic Formula
The Difference of Two Squares
Subject of Formula
Solving Linear Equations
Quadratic Equations
Completing the Square
Sequences and Nth Term
Arithmetic and Geometric Sequences
Quadratic Sequences
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Ratio, Proportion and Rates of Change
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Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
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2D Shapes
2D Shapes Quiz
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3D Shapes Quiz
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
Triangles Quiz
Angles in Triangles
Polygons
Lines of Symmetry
Bearings
Loci and constructions
Constructing Triangles
Transformations
Enlargement
Plans and elevations
Areas and Volumes in Map Scale Problems
Probability
10 Topics
Introduction to Probability
Relative Frequency
Sample Space
Probability of Single Events
Probability of Combined Events
Probability Trees
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
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Population and samples
Representing data
Vectors
Mensuration and Calculation
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Area
Volume
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Probability of Combined Events

GCSE Maths Probability Probability of Combined Events

For some probability questions, you may be asked to calculate the probability of more than one event happening.

Example: Using the ‘AND’ Rule

If I flipped a coin and rolled a dice, what is the probability that I got a head and a 6?

In this case, we would need to work out the probability of each event separately, and then multiply the two fractions together.

Solution:

1. Probability of flipping a head = \frac{1}{2}

2. Probability of rolling a 6 on a 6-sided die: \frac{1}{6}

3. Now, multiply these two fractions together:

\frac{1}{2} \times \frac{1}{6}=\frac{1}{12}

Notice that in this question, we were asked for the probability of the head AND the 6. When you are asked for two events together, you must multiply the fractions.

However, sometimes you may be asked for one event or another event. Here, the keyword is ‘OR’ – this means that we would ADD the fractions, not multiply. Here’s an example:

Example: Using the ‘OR’ Rule

If I flipped a coin and rolled a dice, what is the probability that I get a head or a 6?

As we could have either a head or a 6 (don’t need both together), we would need to work out the probability of each event separately, and then add the two fractions together.

Solution:

1. Probability of flipping a head = \frac{1}{2}

2. Probability of rolling a 6 on a 6-sided dice = \frac{1}{6}

3. Now, add these two fractions together:

\frac{1}{2} + \frac{1}{6} = \frac{4}{6}

= \frac{2}{3}

Summary:

Top tip!

When calculating a probability, the answer must NEVER be greater than 1. If you have added two fractions together and found an answer greater than 1, then it is incorrect.

Independent or Dependent Events

Another part of probability to consider is whether the first event affects the outcome of the second event. For example, consider the probability of the bus being late, and the probability of being late to school.

The second event (being late to school) will depend on the outcome of the first event (whether the bus was late). These are known as dependent events- the probability will change depending on the outcome of the first event.

However, if the first event does NOT affect the second event (for example, the probability of doing your homework, and the probability of being late to school), these are known as independent events.

Example: Dependent Events

Let’s revisit Emma’s buttons again. She has 6 pink, 4 green and 5 blue buttons in a box. She selects two buttons out of the box. What is the probability that they’re both blue?

If Emma is taking two buttons consecutively without replacement, then the second draw depends on the outcome of the first draw.

Therefore, these must be dependent events, as the probability of picking a blue button the second time depends on whether she selected a blue button the first time.

Here’s why:

First event: Selecting a blue button out of the original 15 buttons in the box.

Out of the 15 buttons, there are 5 which are blue. Therefore, the probability of the first button being blue is \frac{5}{15} = \frac{1}{3}

Now that she has taken one blue button, this decreases the total number of buttons by 1, and the number of blue buttons by 1:

Second event: Choosing a second blue button

Now, there are a total of 14 buttons, 4 of which are blue. The probability of her picking a second blue button is now \frac{4}{14}

Combining the probabilities:

The question asks us for the probability of picking two blue buttons. This is the same as selecting a blue AND another blue button. Therefore, we must use the ‘AND’ rule- we must multiply the two fractions.

\frac{1}{3} \times \frac{4}{14} = \frac{4}{42}

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