7.2 Binomial DistributionsThis is a featured page



Binomial Distributions occur when an experiment is repeated and a particular outcome (Success and failure) is counted. Experiments that are repeated are called the Bernoulli Trials.

However, there are certain conditions for a Bernoulli Trial to occur:

  • Only two outcomes are possible (Success and failure)
  • The outcome of each trial does not depend on the previous trial
  • The probability for success and failure is the same for each trial
  • Trials are repeated a specified number of times

An example of when binomial distributions can be used when the number of defects is counted during quality control.

7.2 Binomial Distributions - MDM4U1@FMG



The formula for binomial distributions:
branilio1

Or

binomialcolor

x = Number of single successes
n = Number of Bernoulli Trials
p = Probability of successes outcome
q = Probability of single failure
Also, the sum of the probability of successes and failures always equal one; therefore, p + q = 1.

In addition,
The formula for the expectations for a binomial distribution is:


expected
Example I:
When flipping a coin 6 times, what is the probability of getting 4 heads?

You could make a tree diagram to figure this out but the tree would be difficult to construct. Though if you were to make it, you would see that there are 64 possible outcomes of flipping a coin 6 times and 15 of them give you the event of flipping 4 heads. From which you can find that the probability of flipping 4 heads in 6 flips of a coin is: 15/64

The more efficient way to solve this problem is to use the formula:
binomialcolor

x=4 heads (number of sucesses)
n=6 flips ( number of bernoulli trials)
p=1/2 (probubility of sucesses)
q=1/2 (probibility of failure)

P(4 heads)=6C4*(1/2)^4*(1/2)^2
P(4 heads)=15*(1/16)*(1/4)
P(4 heads)=15/64
You receive the same answer of 15/64 that you would get if you manually made a tree diagram and counted the branches. This example shows why the formula is more efficient.

Example II (indirect method):

What is the probability of tossing at lease two sixes in 6 rolls of a die?

Because the question includes the phrase "at least", it would be highly advisable to use the indirect method.

n = 6 p = 1/6 q = 5/6 x = 0,


binomialcolor
P(x = 0, 1) = 1 - {[6C0 x (1/6)^0 x (5/6)^6 ] + [6C1 x (1/6)^1 x (5/6)^5]}

P(x = 0, 1) = 1 - 0.736775549

P(x = 0, 1) = 0.263224451

Therefore, one has a 26.3% of tossing at least two sixes in 6 rolls of a die


Example III:

For example II, what would the expectations for the binomial distribution?

expected
n = 6 p = 1/6 E(X) = 6(1/6) = 1 If one rolls a die six times, one should expect to get a six only once.

By: Edith, Chrystalin, and Daniel
Edited by: Ayman and Gabriela

References:
Mr. D'Onofrio's lesson sheet

Ms. Richardson's lesson sheet
Data Management Text


Ayman13
Ayman13 		Latest page update: made by Ayman13 , Jun 9 2007, 12:24 AM EDT (about this update About This Update Ayman13 Edited by Ayman13

20 words added
1 word deleted

view changes
- complete history)
Keyword tags: bernoulli trials binomial distruibution Probability
More Info: links to this page

Threads for this page

Started By Thread Subject Replies Last Post
Anonymous Colours 0 Jan 20 2009, 6:34 PM EST by Anonymous
 
Thread started: Jan 20 2009, 6:34 PM EST  Watch
Too many colours... makes it hard to focus
1  out of 5 found this valuable. Do you?    
Keyword tags: None (edit keyword tags)
Mr._D'Onofrio assessment 0 Jun 8 2007, 6:10 PM EDT by Mr._D'Onofrio
Thread started: Jun 8 2007, 6:10 PM EDT  Watch
You did a very good job of improving the clarity of the printed equations. The colours are nice, but perhaps you used too many?

When you wrote the definitions of p and q, you could have written that p is the probability of a SINGLE success, and q is the probability of a SINGLE failure.

Did you or the previous writers spell 'distribution' incorrectly in the tags? Don't worry, this has not reduced you mark. Your comment could have been more specific.

It was nice to see that there were no spelling errors.
1  out of 1 found this valuable. Do you?    
Keyword tags: None (edit keyword tags)
Suuuuu example 0 Jan 10 2007, 6:33 PM EST by Suuuuu
Thread started: Jan 10 2007, 6:33 PM EST  Watch
please further explain your example. nice picture, really eye catching.
0  out of 1 found this valuable. Do you?    
Keyword tags: None (edit keyword tags)

Anonymous  (Get credit for your thread)

Showing 3 of 4 threads for this page - view all

Related Content

  (what's this?Related ContentThanks to keyword tags, links to related pages and threads are added to the bottom of your pages. Up to 15 links are shown, determined by matching tags and by how recently the content was updated; keeping the most current at the top. Share your feedback on Wetpaint Central.)
alia :>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>))))))))))))))))) Mistakes
6.1 Basic Probability Concepts 6.5 Mutually Exclusive Events 6.3 Probabilities Using Counting Techniques