Sample-Space-Diagrams Aqa Higher

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Exam code:8300

Sample space

What is a sample space diagram?

In probability, the sample space means all the possible outcomes
In simple situations it can be given as a list
- For flipping a coin, the sample space is: Heads, Tails
  - the letters H, T can be used
- For rolling a six-sided dice, the sample space is: 1, 2, 3, 4, 5, 6
If there are two sets of outcomes, a grid can be used
- These are called sample space diagrams (or possibility diagrams)
- For example, roll two six-sided dice and add their scores
- A list of all the possibilities would be very long
  - You might miss a possibility
  - It would be hard to spot any patterns in the sample space

Possibility diagram for the sum of scores of two dice

Combining more than two sets of outcomes must be done by listing the possibilities
- For example, flipping three coins
  - The sample space is HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 possible outcomes)

How do I use a sample space diagram to calculate probabilities?

Probabilities can be found by counting the number of possibilities you want, then dividing by the total number of possibilities in the sample space
- In the sample space 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, there are four prime numbers (2, 3, 5 and 7)
  - The probability of getting a prime number is $4 over 10 equals 2 over 5$
- Using the sample space diagram above for rolling two dice, the probability of getting an eight is $5 over 36$
  - There are 5 eights in the grid, out of the total 36 numbers
Be careful, this counting method only works if all possibilities in the sample space are equally likely
- For a fair six-sided dice: 1, 2, 3, 4, 5, 6 are all equally likely
- For a fair (unbiased) coin: H, T are equally likely
- Winning the lottery: Win, Lose are are not equally likely!
  - You cannot count possibilities here to say the probability of winning the lottery is $1 half$

Examiner Tips and Tricks

Some harder questions may not say “by drawing a sample space diagram” so you may have to do it on your own.

Worked Example

Two fair six-sided dice are rolled.

(a) Find the probability that the sum of the numbers showing on the two dice is an odd number greater than 5, giving your answer as a fraction in simplest form.

Draw a sample space diagram to show all the possible outcomes

Circle the possibilities that are odd numbers greater than 5
(5 is not included)

Possibility diagram for the sum of scores of two dice with the odd values greater than 5 circled

Count the number of possibilities that are circled (12) and divide them by the total number of possibilities in the diagram (36)

$12 over 36$

Cancel the fraction

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Maths Gcse Aqa Higher