# How to Assemble Relative Probabilities into a Vector

In quantum physics, probabilities take the place of absolute measurements. Say you've been experimenting with rolling a pair of dice and are trying to figure the relative probability that the dice will show various values. You come up with a list indicating the relative probability of rolling a 2, 3, 4, and so on, all the way up to 12:

Sum of the Dice | Relative Probability (Number of Ways of Rolling a Particular Total) |
---|---|

2 | 1 |

3 | 2 |

4 | 3 |

5 | 4 |

6 | 5 |

7 | 6 |

8 | 5 |

9 | 4 |

10 | 3 |

11 | 2 |

12 | 1 |

In other words, you're twice as likely to roll a 3 than a 2, you're four times as likely to roll a 5 than a 2, and so on. You can assemble these relative probabilities into a vector (if you're thinking of a "vector" from physics, think in terms of a column of the vector's components, not a magnitude and direction) to keep track of them easily:

Okay, now you're getting closer to the way quantum physics works. You have a vector of the probabilities that the dice will occupy various states. However, quantum physics doesn't deal directly with probabilities but rather with *probability amplitudes*, which are the square roots of the probabilities. To find the actual probability that a particle will be in a certain state, you add wave functions of this state — which are going to be represented by these vectors — and then square them. So take the square root of all these entries to get the probability amplitudes:

That's better, but adding the squares of all these should add up to a total probability of 1; as it is now, the sum of the squares of these numbers is 36, so divide each entry by 36^{1/2}, or 6:

So now you can get the probability amplitude of rolling any combination from 2 to 12 by reading down the vector — the probability amplitude of rolling a 2 is 1/6, of rolling a 3 is

and so on.