Probability-Diagrams gcse Edexcel

Sample Space Diagrams

What is a sample space diagram?

• In probability, the sample space means all the possible outcomes

  • A sample space diagram is a way of showing all these outcomes in a systematic and organised way

• In simple situations sample space diagram can just be a list

  • For flipping a coin, the sample space is: Heads, Tails

    • the letters H, T can be used

    • For flipping two coins the sample space could be given as: HH, HT, TH, TT (4 possible outcomes)

  • For rolling a six-sided dice, the sample space is: 1, 2, 3, 4, 5, 6

    • But for rolling two dice there would be 36 possibilities!

• When combining two things a grid can be used to show the sample space

  • For example, rolling two six-sided dice and adding their scores

    • A list of all the possibilities would be very long

      • It would be hard to spot if you had missed any possibilities

      • It would be hard to spot any patterns

  • Use a grid instead

• If you need to combine more than two things you’ll probably need to go back to using a list

  • For example, flipping three coins (or flipping one coin three times!)

    • In this case 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 often be found by counting the possibilities you want,

  • then dividing by the total number of possibilities in the sample space

• For example, in the sample space 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 you can count 4 prime numbers (2, 3, 5 and 7)

  • So the probability of getting a prime number is

• Or for rolling two dice and adding the results, the possibility diagram above shows there are 5 ways to get ‘8’, and 36 outcomes in total

  • So the probability of getting an 8 is

• But 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 coin: H, T are equally likely

  • Winning the lottery: Yes, No. These are not equally likely! 

    • You cannot count possibilities here to say the probability of winning the lottery is  

• This method can also be used for finding the probability of an event occurring given that another event has occurred (conditional probability)

  • For example when two dice are rolled, you can use the sample space diagram above to find the probability that an individual dice shows a 6, given that the total showing on the two dice is 7

    • count the number of outcomes that sum to 7 (there are 6 of them) – this goes in the denominator

    • count the number of those outcomes in which one of the dice shows a 6 (there are two of these, (1,6) and (6,1)) – this goes in the numerator

    • So the probability is