Probability calculus, like all the other mathematical theories, starts with certian primitive notions which are not defined (see 7.6). An example of such notion here is an elementary event. An elementary event is a possible outcome of an experiment. The set of all elementary events in a certain experiment is called a sample space and usually marked with the Ω symbol. The sample space of a random experiment plays the role of the universal set when modeling the experiment. The notions of an elementary event and a sample space will be explained using practical examples.

Example 13.1.1

1. Let the trial be rolling a die. We look at the number of pips at the upper face of a die. Elementary event is receiving a certain number of pips. Obviously, the possible rolls are 1, 2, 3, 4, 5 or 6 pips. Hence the sample space Ω contains, in this trial, 6 events which can be identified with the number of pips Ω = {1, 2, ...,6}.
   
2. Let the trial be a toss of a coin. We look at the top side of a coin and we record a fact whether it is heads or tails. This time there are only two elementary events: heads or tails. We mark those events accordingly H and T. The sample space is in this case a set of 2 elements, Ω = {H, T}.
   
3. Let the trial be a toss of two coins. Just like in the previous example the value of a coin is irrelevant and we only check its upper surface. Elementary events are the ordered pairs (x, y), where x,y ∈ {H,T} and where x indicates the first coin and y - the second one. The sample space consists of four events (H, H), (H, T), (T, H), (T, T).
   

Fig. 13.1.1  Examples of events in a rolling a die trial.

Example 13.1.2

1. During a ski jump competition each contestant jumps twice. The result of each jump is treated as a random event. The length of a jump is measured with an accuracy of 0.5 m. Moreover, we know that the maximum length of a jump is 140 m. The set of elementary events can be identified with the ordered pairs (x,y), where x indicates the length of the first jump and y - the length of the second one. Hence Ω = {(x,y): x length of the first jump, y length of the second jump}. Both x and y can be any of 0, 0.5, 1, 1.5, 2,... 135.5, 140. As there is 281 numbers in this set, | Ω | = 281*281. The sample space contains 78961 events.
   
2. The final grade of a course depends on the sum of points received from two tests and an exam. The maximum number of points to receive is 20 for each test and 60 at the exam. The sample space Ω in this case can be the set of threes (x,y,z), where x, y are the points from the tests and z - from the exam. We have Ω ={(x,y,z) ∈ N³: x ≤ 20, y ≤ 20, z ≤ 60}. This sample space consists of 21*21*61 elements.
   

Question 13.1.1 How many elements are there in the sample space if the trial is rolling 3 dice?

Question 13.1.2 Specify the sample space of a trial which is painting a tricolour flag (three stripes) using three different colours a, b, c.

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