1.- If we throw an eight-sided die which of the following events are equally likely?

a) Throwing a 5
b) Throwing an 8
c) Throwing a 9
d) Throwing an even number
e) Throwing a multiple of 3
f)  Throwing a 4

Work out the theoretical probability of each of these events.

Remember that we work out this probability as follows:

In this window we can simulate throwing the eight-sided die.
Look carefully at which numbers are included in the successful outcomes for each of the probabilities listed above.

This window illustrates the eight sides of the octahedron, which has been opened out. Each time you click on the blue arrow one of the sides of the die lights up indicating the number thrown.

2.- If we throw a twelve-sided die which of the following events are equally likely?

a) Throwing a 1
b) Throwing an even number
c) Throwing a multiple of 4
d) Throwing a 10
e) Throwing a 15
f)  Throwing a 7
g) Throwing a number between 5 and 11

Work out the theoretical probability of each of these events.

3.- If we throw a twenty-sided die which of the following events are equally likely?

a) Throwing a 2
b) Throwing a multiple of 10
c) Throwing a 25
d) Throwing a 13
e) Throwing an odd number
f)  Throwing a number lower than 15
g) Throwing a number greater than 11
h) Throwing a 20
i)  Throwing a number between 10 and 17

Work out the theoretical probability of each of these events.

In this example we are going to refer to a Spanish deck of cards. There are four suits (Oros, Copas, Espadas and Bastos) and each suit contains the following cards in ascending order: Ace, 2, 3, 4, 5, 6, 7, Jack (symbolised by a stick man), Queen (symbolised by a horse!!) and King (symbolised by a crown).

If we SELECT A CARD from a Spanish deck of cards what is the probability of selecting: 
    - The ace of Oros                                     - A Queen
    - The King of Copas                                   - A Basto ?

If we repeated this experiment (of selecting a card) a thousand times how many times would you expect each of the events above to be successful?

If the selected card is the six of espadas, several other successful events have occurred such as "selecting an espada", "selecting a six", "selecting a number lower than 7", etc.

On the other hand, several more events have not been successful, such as "selecting the six of oros", "selecting the seven of espadas", "selecting a copa", etc.

Therefore, some events can occur simultaneously and others cannot.

Can any of the events mentioned above happen at the same time?

Look carefully at the following events:

- Selecting a picture card (jack, queen or king)
- Selecting an oro

If you selected a card a thousand times how often would these two events occur at the same time?

Move points C and D slowly using the mouse to hide cards from left to right or vice versa. Move points A and B similarly to hide cards from top to bottom or vice versa. Drag points C and D horizontally so that only the picture cards are showing. Drag points A and B vertically so that only cards in the oros suit are showing.

Is any card still uncovered after this? Can both events occur at the same time?

Use a similar process to the one above to work out if the following pairs of events can happen at the same time or not:

- Selecting the ace of oros and the king of copas
- Selecting a queen and a basto

To sum up, we can say that two events are mutually exclusive if they cannot occur simultaneously and not mutually exclusive if they can happen at the same time.