Illustration of conditional probability.

S is the sample space, A and B_n are events. 

Overall, P (A) ≈ 0.33. However (A | B₁) = 1, P(A | B₂) ≈ 0.12 and P(A | B₃) = 0

In general, the probability of an event depends on the circumstances in which it occurs. A conditional probability is the probability of an event, assuming a particular set of circumstances. More formally, a conditional probability is the probability that event A occurs when the sample space is limited to event B. This is notated P (A | B), and is read "the probability of A given B".

In the notation P( A | B ) the symbol P is used, only as a reference to the original probability. It should not be read as the probability P of some event A|B. Sometimes the more accurate notation P_B(A) is used, but the use of complex events as index of the symbol P is cumbersome. Notice that the line separating the two events A and B is a vertical line.

Joint probability is the probability of two events in conjunction. That is, it is the probability of both events together. The joint probability of A and B is written  or 

Conditioning of probabilities, i.e. updating them to take account of (possibly new) information, may be achieved through Bayes' theorem. In such conditioning, the probability of A given only initial information I, P(A|I), is known as the prior probability. The updated conditional probability of A, given I and the outcome of the event B, is known as the posterior probability (A|B, I).

Consider the simple scenario of rolling two fair six-sided dice, labeled die 1 and die 2. Define the following three events (not assumed to occur simultaneously): A: Die 1 lands on 3.

B: Die 2 lands on 1.

C: The dice sum to 8.

The prior probability of each event describes how likely the outcome is before the dice are rolled, without any knowledge of the roll's outcome. For example, die 1 is equally likely to fall on each of its 6 sides, so P (A) = 1/6. Similarly P (B) = 1/6. Likewise, of the 6 × 6 = 36 possible ways that a pair of dice can land, just 5 result in a sum of 8 (namely 2 and 6, 3 and 5, 4 and 4, 5 and 3, 6 and 2), so P(C) = 5/36.

Some of these events can both occur at the same time; for example events A and C can happen at the same time, in the case where die 1 lands on 3 and die 2 lands on 5. This is the only one of the 36 outcomes where both A and C occur, so its probability is 1/36. The probability of both A and C occurring is called Some of these events can both occur at the same time; for example events A and C can happen at the same time, in the case where die 1 lands on 3 and die 2 lands on 5. This is the only one of the 36 outcomes where both A and C occur, so its probability is 1/36. The probability of both A and C occurring is called  the joint probabilityof A and C and is written , so . On the other hand, if die 2 lands on 1, the dice cannot sum to 8, so .

Now suppose we roll the dice and cover up die 2, so we can only see die 1, and observe that die 1 landed on 3. Given this partial information, the probability that the dice sum to 8 is no longer 5/36; instead it is 1/6, since die 2 must land on 5 to achieve this result. This is called the conditional probability, because it is the probability of C under the condition that A is observed, and is written P(C | A), which is read "the probability of C given A." Similarly, P(C | B) = 0, since if we observe die 2 landed on 1, we already know the dice can't sum to 8, regardless of what the other die landed on.

On the other hand, if we roll the dice and cover up die 2, and observe die 1, this has no impact on the probability of event B, which only depends on die 2. We say events A and B are statistically independent  or just independent and in this case

In other words, the probability of B occurring after observing that die 1 landed on 3 is the same as before we observed die 1.

Intersection events and conditional events are related by the formula:

(Our solved example in mathguru.com uses this concept)

In this example, we have:

As noted above, , so by this formula:

On multiplying across by P(A),

In other words, if two events are independent, their joint probability is the product of the prior probabilities of each event occurring by itself.

http://en.wikipedia.org/wiki/Conditional_probability

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