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Further Maths

  • A-Level Maths

Probability

Probability is used to predict the likeliness of something happening. It is always given between 0 and 1.

  • An experiment is a repeatable process that can have a number of outcomes

  • An event is one single or multiple outcomes

  • A sample space is the set of all possible outcomes

For two events, E₁ and E₂, with probabilities P₁ and P₂ respectively:

  • To find the probability of either E₁ or E₂ happening, add the two probabilities, P = P₁ + P₂

  • To find the probability of both E₁ and E₂ happening, multiply the two probabilities, P = P₁ x P₂

The sample space for rolling two fair six-sided dice and adding up the numbers that show would look like this:

To work out the probability of getting a particular result, you count how many times the result occurs and divide by the total number, 36 (since 6² = 36). So to work out the probability of getting a 10, count the number of tens and divide by 36: 3/36 = 0.0833

Generally, give your answers as decimals to three significant figures


Conditional Probability

If the probability of an event is dependent on the outcome of the previous event, it is called conditional. Conditional probability is noted using a vertical line between the events:

The probability of B occurring, given that A has already occurred is given by P(B|A)

For two independent events:

P(A|B) = P(A|B') = P(A)

Experiments with conditional probability can be calculated using a two-way table/restricted sample space:



Venn Diagrams

A Venn diagram is used to represent events happening. The rectangle represents the sample space, and the subsets within it represent certain events. Set notation is used to describe events within a sample space:

A B represents the intersection
A ∪ B represents the union
Adding a dash, ', means the compliment, or "not"

The addition formula is very useful:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Mutually Exclusive Events

As you can see from the Venn diagrams, mutually exclusive events do not intersect. This means there is no overlap, so:

P(A or B) = P(A) + P(B)

Independent Events

When one event has no effect on the other, the two events are described as independent. Therefore:

P(A ∪ B) = P(A) x P(B)

Conditional Probability in Venn Diagrams

You can find conditional probability easily from Venn diagrams using the multiplication formula:

P(B|A) = P(B ∩ A) / P(A)


Tree Diagrams

Tree diagrams are used to show the outcomes of two or more events happening, one after the other.


For example, if there are 3 red tokens and 7 blue tokens in a bag, and two are chosen one after the other without replacement (the first is not put back into the bag), a tree diagram can model this:

When you have worked out the probability of each branch, add them together - if they sum to 1, it is correct.



Conditional Probability in tree diagrams

Tree diagrams show conditional probability in their second and third etc columns.

The multiplication formula still applies:

P(B|A) = P(B ∩ A) / P(A)
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