UK Secondary (7-11)
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Theoretical Probability

Sample Space

The sample space, sometimes called and event space,  is a listing of all the possible outcomes that could arise from an experiment. 

For example

  • tossing a coin would have a sample space of {Head, Tail}, or {H,T}
  • rolling a dice would have a sample space of {1,2,3,4,5,6}
  • watching the weather could have a sample space of {sunny, cloudy, rainy} or {hot, cold}
  • asking questions in a survey of favourite seasons could have a sample space of {Summer, Autumn, Winter, Spring}

Did you also notice how I listed the sample space?  Using curly brackets { }.


An event is the word used to describe a single result from within the sample space.  It helps us to identify which of the sample space outcomes we might be interested in.  

For example, these are all events.

  • Getting a tail when a coin is tossed.  
  • Rolling more than 3 when a dice is rolled
  • Getting an ACE when a card is pulled from a deck 

We use the notation, P(event) to describe the probability of particular events.  

Adding up how many times an event occurred during an experiment gives us the frequency of that event.  

The relative frequency is how often the event occurs compared to all possible events and is also known as the probability of that event occurring.

Probability Values

The probability values that events can take on range between 0 (impossible) and 1 (certain).


We can calculate probabilities by constructing a fraction like this: $\frac{\text{what you want}}{\text{total }}$what you wanttotal  which we write more formally as

P(event) = $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes



Worked Examples


The $26$26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is to be picked out of the bag at random what is the probability of picking a:

  1. J?

  2. K, Y or R?

  3. Letter in the word PROBABILITY?

  4. M, K, D, O, H or B?

  5. Letter in the word WORKBOOK?


A cube has six faces. Each face is painted a certain colour and the cube is rolled. How many faces should be painted blue so that the probability of blue appearing on the uppermost face is:

  1. $\frac{1}{2}$12?

  2. $\frac{1}{3}$13?

  3. $\frac{1}{6}$16?

  4. $1$1?


A book has pages numbered from $1$1 to $100$100. If the book is opened to a random page, what is the probability that the page number:

  1. Is a multiple of $9$9?

  2. Has digit ‘$6$6’ in the page number?

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