Probabilities are numbers in the [0, 1] interval, that is, numbers between 0 and 1, including both extremes. Probabilities follow a few rules; one of these rules is the product rule:

We read this as follows: the probability of  and  is equal to the probability of  given , times the probability of . The  expression represents the joint probability of  and . The  expression is used to indicate a conditional probability; the name refers to the fact that the probability of  is conditioned on knowing . For example, the probability that the pavement is wet is different from the probability that the pavement is wet if we know (or given that) it's raining. A conditional probability can be larger than, smaller than, or equal to the unconditioned probability. If knowing  does not provides us with information about , then . This will be true only if  and  are independent of each other. On the contrary, if knowing  gives us useful information about , then the conditional probability could be larger or smaller than the unconditional probability, depending on whether knowing  makes  less or more likely. Let's see a simple example using a fair six-sided die. What is the probability of getting the number 3 if we roll the die, ? This is 1/6 since each of the six numbers has the same chance for a fair six-sided die. And what is the probability of getting the number 3 given that we have obtained an odd number, ? This is 1/3, because if we know we have an odd number, the only possible numbers are  and each of them has the same chance. Finally, what is the probability of ? This is 0, because if we know the number is even, then the only possible ones are , and thus getting a 3 is not possible.

As we can see from this simple example, by conditioning on observed data we effectively change the probability of events, and with it, the uncertainty we have about them. Conditional probabilities are the heart of statistics, irrespective of your problem being rolling dice or building self-driving cars.