Tool for performing probability calculations with the binomial distribution, number of k successes, average odds, etc.
Binomial Distribution - dCode
Tag(s) : Combinatorics
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Binomial Distribution
Probability of K Successes
Answers to Questions (FAQ)
What is the binomial distribution? (Definition)
The binomial distribution is a model (a law of probability) which allows a representation of the average number of successes (or failures) obtained with a repetition of successive independent trials.
$$ P(X=k) = {n \choose k} \, p^{k} (1-p)^{n-k} $$
with $ k $ the number of successes, $ n $ the total number of trials/attempts/expériences, and $ p $ the probability of success (and therefore $ 1-p $ the probability of failure).
When to use the binomial distribution?
The binomial distribution can be used in situations with 2 contingencies (success or failure, true or false, toss or tails, etc.) that can be repeated and independent.
Example: Calculation of the probability to draw 4 times the number 6 after 5 successive dice rolls: the probability $ p $ to make a 6 is $ 1/6 $, the total number of trials is $ n = 5 $, the total number of successes expected is $ k = 4 $. $$ P(X=4) = {5 \choose 4} \, \left(\frac{1}{6}\right)^4 \left(1-\frac{1}{6}\right)^{5-4} = {5 \choose 4} \left(\frac{1}{6}\right)^4 \left(\frac{5}{6}\right)^1 = \frac{5^2}{6^5} \approx 0.00321 \approx 0.3% $$
Exercises involving the binomial distribution look like:
— A person randomly draws a card 10 times from a deck of 52 cards. What is the probability that she draws at least 7 red color cards?
— A basketball player hits 60% of his free throws. If the player attempts 10 free throws, what is the probability that he will hit exactly 8?
— A merchant has a conversion rate of 10% on his online sales site. If 200 people visit his site, what is the probability that 50% will make a purchase?
— A medicine has a 75% success rate in relieving an illness. If given to 10 patients, what is the probability that at least 8 patients will feel better after taking the drug?
Why is this law called binomial?
The formula for the binomial distribution involves the binomial coefficient $ {n \choose k} $ (which can be read as a combination of $ k $ among $ n $).
It is sometimes called Two-outcome distribution and is closely related to Bernoulli distribution.
Source code
dCode retains ownership of the "Binomial Distribution" source code. Except explicit open source licence (indicated Creative Commons / free), the "Binomial Distribution" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Binomial Distribution" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Binomial Distribution" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
The copy-paste of the page "Binomial Distribution" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Binomial Distribution on dCode.fr [online website], retrieved on 2024-11-21,
