Explanation:
Conditional probability
Illustration
of conditional probability.
S is the sample space, A and B_{n} are events.
Overall, P (A)
≈ 0.33. However (A | B_{1}) = 1, P(A | B_{2}) ≈ 0.12 and P(A | B_{3}) = 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".
Notation
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.
Terminology
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).
Introduction
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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