How do you weight probabilities to add up to 1?
Example:
Chance of rain .25
Chance of person wearing purple hat .1
Chance of person wearing red hat .05
Chance of person wearing black hat .2
Chance of person wearing yellow hat .1
Chance of person wearing white hat .3
So, You run into a person wearing a hat in the rain.
The chance of running into people in each color of hat in the rain is
Chance of person wearing purple hat .025
Chance of person wearing red hat .0125
Chance of person wearing black hat .05
Chance of person wearing yellow hat .025
Chance of person wearing white hat .075
Would I weight this to make it all add up to 1? What is what I need to do called?
To weight probabilities so that they add up to 1, you need to normalize them. Normalization is the process of dividing each probability by the sum of all probabilities.
In your example, you have probabilities for the chance of rain and the chance of a person wearing a hat in different colors. To find the probability of running into people in each color of hat in the rain, you multiply the probability of each event occurring by the probability of rain. This is known as conditional probability.
To normalize these conditional probabilities, you divide each of them by the sum of all the conditional probabilities. In your case, the sum of the conditional probabilities is:
Chance of person wearing purple hat in the rain + Chance of person wearing red hat in the rain + Chance of person wearing black hat in the rain + Chance of person wearing yellow hat in the rain + Chance of person wearing white hat in the rain
= 0.025 + 0.0125 + 0.05 + 0.025 + 0.075
= 0.1875
Then, you divide each of the conditional probabilities by 0.1875 to normalize them:
Chance of person wearing purple hat in the rain = 0.025 / 0.1875 = 0.1333
Chance of person wearing red hat in the rain = 0.0125 / 0.1875 = 0.0667
Chance of person wearing black hat in the rain = 0.05 / 0.1875 = 0.2667
Chance of person wearing yellow hat in the rain = 0.025 / 0.1875 = 0.1333
Chance of person wearing white hat in the rain = 0.075 / 0.1875 = 0.4
Now, the probabilities of running into people in each color of hat in the rain add up to 1:
0.1333 + 0.0667 + 0.2667 + 0.1333 + 0.4 = 1
This process of dividing each probability by the sum of all probabilities to make them add up to 1 is called normalization.