Conditional probability

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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.