Combined Probabilities (Edexcel GCSE Maths)

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Written by: Mark Curtis

Reviewed by: Dan Finlay

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Combined Probability

How do I calculate combined probabilities?

You can calculate probabilities of one event after another without needing tree diagrams
- These are called combined (or successive) probabilities
There are two rules to learn
- And means multiply and or means add
- P(A and B) = P(A) x P(B)
- P(AA or BB) = P(AA) + P(BB)
Try to rephrase each question using and / or
- For example, when flipping a coin twice:
  - P(two heads) = P(head and head)
  - P(both the same) = P(head and head or tail and tail) = P(HH) + P(TT)
Remember that P(not A) = 1 - P(A)

Worked Example

A box contains 3 blue counters and 8 red counters.
A counter is taken at random and its colour is noted.
The counter is put back into the box.
A second counter is then taken at random, and its colour is noted.

Work out the probability that

(a) both counters are red,

P(both red) = P(red and red)
This is P(red) × P(red) using the 'and rule'

$\begin{array}{rcl} \frac{8}{11} \times \frac{8}{11} \end{array}$

$\begin{array}{rcl} \frac{64}{121} \end{array}$

(b) the two counters are of different colours.

P(different colours) = P(blue and red or red and blue)
This is P(B and R) + P(R and B) using the 'or rule'
This is P(B) × P(R) + P(R) × P(B) using the 'and rule' twice

$\begin{array}{rcl} \frac{8}{11} \times \frac{3}{11} + \frac{3}{11} \times \frac{8}{11} \\ = & \frac{24}{121} + \frac{24}{121} \end{array}$

$\begin{array}{rcl} \frac{48}{121} \end{array}$

Last updated: 18 October 2024

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