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When you flip a coin, there are two possible outcomes: heads and tails. Each outcome has a fixed probability, the same from trialto trial. In the case of coins, heads and tails each have the same probability of 1/2. More generally, there are situations inwhich the coin is biased, so that heads and tails have different probabilities. In the present section, we consider probabilitydistributions for which there are just two possible outcomes with fixed probability summing to one. These distributions arecalled are called binomial distributions .

A simple example

The four possible outcomes that could occur if you flipped a coin twice are listed in [link] . Note that the four outcomes are equally likely: each has probability 1 4 . To see this, note that the tosses of the coin are independent(neither affects the other). Hence, the probability of a head on Flip 1 and a head on Flip 2 is the product of P[H] and P[H] , which is 1 2 1 2 1 4 . The same calculation applies to the probability of a head on Flip one and a tail on Flip 2. Each is 1 2 1 2 1 4 .

Four possible outcomes
Outcome First Flip Second Flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails

The four possible outcomes can be classifid in terms of the number of heads that come up. The number could betwo (Outcome 1), one (Outcomes 2 and 3) or 0 (Outcome 4). The probabilities of these possibilities are shown in [link] and in [link] . Since two of the outcomes represent the case in which just one head appears inthe two tosses, the probability of this event is equal to 1 4 1 4 1 2 . [link] summarizes the situation.

Probabilities of getting 0,1, or 2 heads.
Number ofHeads Probability
0 1/4
1 1/2
2 1/4
Probabilities of 0, 1, and 2 heads.

[link] is a discrete probability distribution: It shows the probability for each of the values on theX-axis. Defining a head as a "success," [link] shows the probability of 0, 1, and 2 successes for two trials (flips) for an event that has a probability of 0.5 of being asuccess on each trial. This makes [link] an example of a binomial distribution .

The formula for binomial probabilities

The binomial distribution consists of the probabilities of each of the possible numbers of successes on n trials for independent events that each have a probability of p of occurring. For the coin flip example, n 2 and p= 0.5 . The formula for the binomial distribution is shown below: P[x] n x n x p x 1 p n x where P[x] is the probability of x successes out of n trials, n is the number of trials, and p is the probability of success on a given trial. Applying this to thecoin flip example, P[0] 2 0 2 0 0.5 0 1 0.5 2 0 2 2 1 .25 0.25 P[1] 2 1 2 1 0.5 1 1 0.5 2 1 2 1 .5 .5 0.50 P[2] 2 2 2 2 0.5 2 1 0.5 2 2 2 2 .25 1 0.25 If you flip a coin twice, what is the probability of getting one or more heads? Since the probability of getting exactlyone head is 0.50 and the probability of getting exactly two heads is 0.25, the probability of getting one or more heads is 0.50 0.25 0.75 .

Now suppose that the coin is biased; let's say the probability of heads is only 0.4. What is the probability of getting heads at leastonce in two tosses? We could substitute p=0.4 with x=1 and with x=2 into our general formula above; adding the results would obtain the answer 0.64.

Cumulative probabilities

We toss a coin 12 times. What is the probability that we get from 0 to 3 heads? The answer is found by computing theprobability of exactly 0 heads, exactly 1 head, exactly 2 heads, and exactly 3 heads. The probability of getting from 0to 3 heads is then the sum of these probabilities. The probabilities are: 0.0002, 0.0029, 0.0161, and 0.0537. The sumof the probabilities is 0.073. The calculation of cumulative binomial probabilities can be quite tedious. Therefore we haveprovided a binomial calculator to make it easy to calculate these probabilities.

Click here for the binomial calculator.

Mean and standard deviation of binomial distributions

Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads. If you performedthis experiment over and over again, what would the mean number of heads be? On average, you would expect half the cointosses to come up heads. Therefore the mean number of heads would be 6. In general, the mean of a binomial distributionwith parameters n (the number of trials) and p (the probability of success for each trial) is: μ n p where μ is the mean of the binomial distribution. The variance of the binomial distribution is: σ 2 n p 1 p where σ 2 is the variance of the binomial distribution.

Let's return to the coin tossing experiment. The coin was tossed 12 times so n 12 . A coin has a probability of 0.5 of coming up heads. Therefore, p 0.5 . The mean and standard deviation can therefore be computed as follows: μ n p 12 0.5 6 σ 2 n p 1 p 12 0.5 1.0 0.5 3.0 Naturally, the standard deviation σ is the square root of the variance σ 2 .

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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