Search for a tool
Binomial Distribution

Tool for performing probability calculations with the binomial distribution, number of k successes, average odds, etc.

Results

Binomial Distribution -

Tag(s) : Combinatorics

Share dCode and more

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!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Binomial Distribution' tool, so feel free to write! Thank you!

# Binomial Distribution

## Probability of K Successes

### (aka) Mass Function

 Probability Format Numeric (between 0 and 1) Percentage (in %) Fraction A/B (exact or approximate) Odds (About ≈) X in N Chance

### What is the binomial distribution? (Definition)

The binomial distribution is a model which allows a representation of the average number of successes 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 $k = 4$. $$P(X=4) = {5 \choose 4} \, \left(\frac{1}{6}\right)^4 \left(1-\frac{1}{6}\right)^{5-4} = {5 \choose 4} \left(\frac{1}{6}\right)^4 \left(\frac{5}{6}\right)^1 = \frac{5^2}{6^5} \approx 0.00321 \approx 0.3%$$

### Why is the binomial 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 $n$).

## Source code

dCode retains ownership of the online "Binomial Distribution" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Binomial Distribution" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, 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, copy-paste, or API access for "Binomial Distribution" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!