Explanation:
Independence (probability
theory)
In probability
theory, to say that two events are independent intuitively means that the occurrence of one event
makes it neither more nor less probable that the other occurs. For example:
1. The
event of getting a 6 the first time a die is rolled and the event of getting a
6 the second time are independent.
2. By
contrast, the event of getting a 6 the first time a die is rolled and the event
that the sum of the numbers seen on the first and second trials is 8 are not independent.
3. If two
cards are drawn with replacement from a deck of cards, the
event of drawing a red card on the first trial and that of drawing a red card
on the second trial are independent.
4. By
contrast, if two cards are drawn without replacement from a
deck of cards, the event of drawing a red card on the first trial and that of
drawing a red card on the second trial are again not independent.
Similarly, two random
variables are independent if the
conditional probability distribution of either given the observed value of the
other is the same as if the other's value had not been observed. The concept of
independence extends to dealing with collections of more than two events or
random variables.
In some instances, the term "independent" is replaced by
"statistically independent", "marginally independent",
or "absolutely independent".
The standard
definition says:
Two events A and B are independent if and only if Pr(A ∩ B) = Pr(A)Pr(B).
Here A ∩ B is the intersection of A and B, that is, it is the event that
both events A and B occur.
More generally, any collection of
events-possibly more than just two of them-are mutually independent if
and only if for every finite subset A_{1}, ..., A_{n} of
the collection we have
This is called the multiplication
rule for independent events. Notice that independence requires this
rule to hold for every subset of the collection; see for
a three-event example in which and
yet no two of the three events are pairwise independent.
http://en.wikipedia.org/wiki/Independence_(probability_theory)#Independent_events
Bernoulli trial
In the theory of probability and statistics,
a Bernoulli trial is an experiment whose outcome is
random and can be either of two possible outcomes, "success" and
"failure".
In practice it refers to a single experiment which can have one of two possible
outcomes. These events can be phrased into "yes or no" questions:
1. Did the
coin land heads?
2. Was the
newborn child a girl?
Therefore success and failure are labels
for outcomes, and should not be construed literally. Examples of Bernoulli
trials include
3. Flipping
a coin. In this context, obverse ("heads") conventionally denotes
success and reverse ("tails") denotes failure. A fair coin has the
probability of success 0.5 by definition.
4. Rolling
a die, where a six is "success" and everything else a
"failure".
5. In
conducting a political opinion poll, choosing a voter at random to ascertain
whether that voter will vote "yes" in an upcoming referendum.
Independent repeated trials of an
experiment with two outcomes only are called Bernoulli trials. Call one of the
outcomes success and the other outcome failure. Let p be
the probability of success in a Bernoulli trial. Then the probability of
failure q is given by
q =
1 − p.
A binomial experiment consisting of a
fixed number n of trials, each with a probability of success p, is
denoted by B(n,p). The probability of k success
in the experiment B(n,p) is
given by:
.
The function P(k) for for B(n,p) is
called a binomial distribution. (Our solved example in mathguru.com uses this concept)
Bernoulli trials may also lead to negative
binomial, geometric, and other distributions as well.
http://en.wikipedia.org/wiki/Bernoulli_trial
The above explanation is copied from
Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons
Attribution- ShareAlike 3.0 Unported License.