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Binomial Distribution

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

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Binomial Distribution -

Tag(s) : Combinatorics

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# 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 (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.

### What are the properties of the binomial distribution?

The expectation (the average of the positive results) is equal to $E(X) = n \times p$

The variance is equal to $V(X) = n p (1-p)$

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Binomial Distribution on dCode.fr [online website], retrieved on 2023-02-08, https://www.dcode.fr/binomial-distribution

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