Conditional probability

Conditional probability is one of the most important concepts in probability theory, indicating the probability if one or more conditions are met. In theory it is the probability that a given event occurs given the knowledge of some partial information about the results of the experiment. As a result of this knowledge the possible outcomes may be restricted, and hence the probability of any event may change. This changed probability is the conditional probability.

For example, take a die tossing experiment. Assuming the die is fair, the probability of it falling on 1, 2, 3, 4, 5 or 6 is 1/6 (evenly split). If we are given partial information about the final result e.g. The die falls on an even number (i.e either 2, 4 or 6), the conditional probabilities for all 6 faces of the die change. The probability of obtaining a 1, 3 or 5 will go down to 0, while the probability of obtaining a 2, 3 or 6 will go up to 2/6 (or 1/3). These new probabilities are conditioned on the fact that our result is even, and therefore called conditional probabilities.

As another example consider an urn with 10 balls numbered from 0 to 9. Balls are draw randomly without replacement. The probability to get ball number 0 in the first draw is evidently 1/10. But also for the second and 10th draw is the probability 1/10 to get ball number 0. However if the first draw resulted in ball 9 having been drawn, there is only a chance of 1/9 to get number 0. This chance is the conditional probability given ball number 9 was first drawn.