Random experiments  (Page 2/3)

There are three important concepts associated with a random experiment: 'outcome', 'sample space', and 'event'. Two examples of experiments will be used to familiarize you with these terms:

• In Experiment 1 a single die is thrown and the value of the top face after it has come to rest is noted.
• In Experiment 2 two dice are thrown at the same time and the total of the values of each of the top faces after they have come to rest is noted.

Outcome

An outcome of an experiment is a single result of that experiment.

• A possible outcome of Experiment 1: the value on the top face is '3'
• A possible outcome of Experiment 2: the total value on the top faces is '9'

Sample space

The sample space of an experiment is the complete set of possible outcomes of the experiment.

• Experiment 1: the sample space is 1,2,3,4,5,6
• Experiment 2: the sample space is 2,3,4,5,6,7,8,9,10,11,12

Event

An event is any set of outcomes of an experiment. (You can think of it as 'the outcomes we are looking for' or favourable outcomes.)

• A possible event of Experiment 1: an even number being on the top face of the die
• A possible event of Experiment 2: the numbers on the top face of each die being equal

A Venn diagram can be used to show the relationship between the possible outcomes of a random experiment and the sample space. The Venn diagram in [link] shows the difference between the universal set, a sample space and events and outcomes as subsets of the sample space.

Venn diagrams can also be used to indicate the union and intersection between events in a sample space ( [link] ).

Complement

A final notion that is important to understand is the notion of complement . Just as in geometry when two angles were called 'complementary' if they added up to 180 degrees, (the two angles 'complement' each other to make a 'whole' straight line), the complement of a set of outcomes $S$ , usually denoted $S^c$ is the set of all outcomes in the sample space but not in $S$ (i.e., $S^c$ complements $S$ to form the entire sample space). Thus, by definition, $S\cup S^c=S$ is always true. So in the Exercise above, the complement of P (i.e. P^c) = {1,4,6,8,9}, while E^c = {1,3,5,7,9}. So n(P^c) = n(E^c) = 5

In theory, it is very easy to calculate complements, since the number of elements in the complement of a set is just the total number of outcomes in the sample space minus the outcomes in that set (in the example above, there were 9 possible outcomes in the sample space, and 4 possible outcomes in each of the sets we were interested in, thus both complements contained 9-4 = 5 elements). Similarly, it is easy to calculate probabilities of complements of events since they are simply the total probability (e.g. 1 if our total measure is 1) minus the probability of the event in question.

Random experiments

1. Let $S$ denote the set of whole numbers from 1 to 16, $X$ denote the set of even numbers from 1 to 16 and $Y$ denote the set of prime numbers from 1 to 16
   1. Draw a Venn diagram accurately depicting $S$ , $X$ and $Y$ .
   2. Find $n\left(S\right)$ , $n\left(X\right)$ , $n\left(Y\right)$ , $n(X\cup Y)$ , $n(X\cap Y)$ .
   Click here for the solution.
2. There are 79 Grade 10 learners at school. All of these take either Maths, Geography or History. The number who take Geography is 41, those who take History is 36, and 30 take Maths. The number who take Maths and History is 16; the number who take Geography and History is 6, and there are 8 who take Maths only and 16 who take only History.
   1. Draw a Venn diagram to illustrate all this information.
   2. How many learners take Maths and Geography but not History?
   3. How many learners take Geography only?
   4. How many learners take all three subjects?
   Click here for the solution.
3. Pieces of paper labelled with the numbers 1 to 12 are placed in a box and the box is shaken. One piece of paper is taken out and then replaced.
   1. What is the sample space, $S$ ?
   2. Write down the set $A$ , representing the event of taking a piece of paper labelled with a factor of 12.
   3. Write down the set $B$ , representing the event of taking a piece of paper labelled with a prime number.
   4. Represent $A$ , $B$ and $S$ by means of a Venn diagram.
   5. Find
      1. $n\left(S\right)$
      2. $n\left(A\right)$
      3. $n\left(B\right)$
      4. $n(A\cap B)$
      5. $n(A\cup B)$
   6. Is $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$ ?
   Click here for the solution.