# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Binomial distribution

See binomial (disambiguation) for a list of other topics using that name.

In probability theory and statistics, the binomial distribution is a discrete probability distribution which describes the number of successes in a sequence of n independent yes/no experiments, each of which yielding success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. The binomial distribution is the basis for the popular binomial test of statistical significance.

A typical example is the following: assume 5% of the population is HIV-positive. You pick 500 people randomly. How likely is it that you get 30 or more HIV-positives? The number of HIV-positives you pick is a random variable X which follows a binomial distribution with n = 500 and p = .05. We are interested in the probability Pr[X ≥ 30].

In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(n, p). The probability of getting exactly k successes is given by

$P[X=k]={n\choose k}p^k(1-p)^{n-k}\!$

for $k=0,1,2,\dots,n$ and where

${n\choose k}=\frac{n!}{k!(n-k)!}$

is the binomial coefficient "n choose k" (also denoted C(n, k)), whence the name of the distribution. The formula can be understood as follows: we want k successes (pk) and nk failures ((1 − p)nk). However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials.

The cumulative distribution function can be expressed in terms of the regularized incomplete beta function, as follows:

F(k) = I1 - p(n - k,k + 1).

If X ~ B(n, p), then the expected value of X is

E[X] = np

and the variance is

var(X) = np(1 - p).

The most likely value or mode of X is given by the largest integer less than or equal to (n+1)p; if m = (n+1)p is itself an integer, then m − 1 and m are both modes.

If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables, then X + Y is again a binomial variable; its distribution is

B(n + m,p).

Two other important distributions arise as approximations of binomial distributions:

Binomial PDF and Normal approximation for n=6 and p=0.5.
N(np,np(1 - p)).
This approximation is a huge time-saver; historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in 1733. Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed 0-1 indicator variables. Warning: this approximation gives inaccurate results unless a continuity correction is used. Note: that the picture gives the normal and binomial probability density functions (PDF) and not the cumulative distribution functions.
For example, suppose you randomly sample n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If you sampled groups of n people repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation σ = (p(1 − p)/n)1/2. Large sample sizes n are good because the standard deviation gets smaller, which allows a more precise estimate of the unknown parameter p.
• If n is large and p is small, so that np is of moderate size, then the Poisson distribution with parameter λ = np is a good approximation to B(n, p).

The formula for Bézier curves was inspired by the binomial distribution.

## See also

03-10-2013 05:06:04
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
Science kits, science lessons, science toys, maths toys, hobby kits, science games and books - these are some of many products that can help give your kid an edge in their science fair projects, and develop a tremendous interest in the study of science. When shopping for a science kit or other supplies, make sure that you carefully review the features and quality of the products. Compare prices by going to several online stores. Read product reviews online or refer to magazines.

Start by looking for your science kit review or science toy review. Compare prices but remember, Price \$ is not everything. Quality does matter.
 Science Fair Coach What do science fair judges look out for? ScienceHound Science Fair Projects for students of all ages
 All Science Fair Projects.com Site All Science Fair Projects Homepage Search | Browse | Links | From-our-Editor | Books | Help | Contact | Privacy | Disclaimer | Copyright Notice