Tool for performing probability calculations with the binomial distribution, number of k successes, average odds, etc.
Binomial Distribution - dCode
Tag(s) : Combinatorics
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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.
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