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Example:Finding Probability using Binomial Distribution

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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".


Independent events


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 A1, ..., An 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.




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.




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.