## Space Diagrams

A space diagram is a table which displays all the possible outcomes. They are usually used for discrete distributions and are especially useful when 2 or more trials are carried out simultaneously and the results must be combined in some way. For example, the phase space diagrams could show all the possible outcomes if we throw two dice, and the scores on the die are multiplied to obtain the total score. We obtain the space diagram.

| Dice 1 | ||||||

Dice 2 | | 1 | 2 | 3 | 4 | 5 | 6 |

1 | 1 | 2 | 3 | 4 | 5 | 6 | |

2 | 2 | 4 | 6 | 8 | 10 | 12 | |

3 | 3 | 6 | 9 | 12 | 15 | 18 | |

4 | 4 | 8 | 12 | 16 | 20 | 24 | |

5 | 5 | 10 | 15 | 20 | 25 | 30 | |

6 | 6 | 12 | 18 | 24 | 30 | 36 |

Having drawn the diagram it is easy to find probabilities by inspection:for example. If instead we added the scores we would obtain:

| Dice 1 | ||||||

Dice 2 | | 1 | 2 | 3 | 4 | 5 | 6 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | |

2 | 3 | 4 | 5 | 6 | 7 | 8 | |

3 | 4 | 5 | 6 | 7 | 8 | 9 | |

4 | 5 | 6 | 7 | 8 | 9 | 10 | |

5 | 6 | 7 | 8 | 9 | 10 | 11 | |

6 | 7 | 8 | 9 | 10 | 11 | 12 |

Now for example,