# How do you combine probabilities?

Feb 16, 2017

This is quite a complex question, but here is a short explanation.

Let's us a pack of cards as an example. Single event probabilities involve only ONE aspect.

$P \left(\text{Heart}\right) = \frac{13}{52} = \frac{1}{4}$

$P \left(\text{Jack}\right) = \frac{4}{52} = \frac{1}{13}$

If we combine probabilities, we are considering two or more aspects.
If we ask, "What is the probability that a card is a Heart OR a jack, we are widening the options. "OR" implies ADD.

$P \left(H \text{ or } J\right) = P \left(H\right) + P \left(J\right) - P \left(H J\right)$

$= \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$

Note that we subtract the combination that is the Heart and the Jack, because that card has been counted twice.

If we ask, "What is the probability that a card is a Heart AND a jack, we are narrowing the options because we are being more specific. "AND" implies MULTIPLY.

$P \left(H \text{ and } J\right) = P \left(H\right) \times P \left(J\right)$

$= \frac{13}{52} \times \frac{4}{52} = \frac{1}{52}$

The answer makes sense, because there is only one card that is a Heart and a Jack.