Sample space diagrams
Sample space is a term used in mathematics to mean all possible outcomes. For example, the sample space for rolling a normal dice is {1,2,3,4,5,6} as these are all the only outcomes we can obtain.
The sample space for flipping a coin is {H, T}.
What if we wanted to know the possible outcomes for flipping a coin and rolling a dice? The sample space for these two combined events is {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
We could also write out the sample space for rolling two dice, but to simplify things mathematicians often use sample space diagrams.
Look at this sample space diagram for rolling two dice:
From the diagram, we can see that there are 36 possible outcomes. The probability of getting the outcome 3,2 is \(\frac{1}{36}\) because 3,2 only appears once in the sample space diagram and there are 36 outcomes in total.
The probability of getting a 3 and a 2 is \(\frac{2}{36}~=~\frac{1}{18}\). This is because to get a 3 and a 2 we can have either the outcome 3,2 or the outcome 2,3.
Question
A restaurant offers a set menu with a choice of three starters - soup (S), prawn cocktail (P) or bruschetta (B), and three main courses - lamb (L), hake (H) or chicken (C).
Draw a sample space diagram for all the possible combinations of meals that customers of the restaurant could order.
Remember - we cannot say that the probability of a customer ordering soup and lamb is \(\frac{1}{9}\) because each different menu option may not have the same probability of being selected, unlike with rolling two dice or flipping a coin.