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Question: A marksman hits his target with a 20% chance. With what probability does he score more than 9 hits in 11 shots?

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A Bernoulli experiment is a random experiment in which there are only two outputs: Event A occurs or not. If a Bernoulli experiment is carried out n times in a row under the same condition, it is called a Bernoulli chain of the length n.

The occurrence of event A is often referred to as a success or failure. Therefore, the probability P(A) =p is also called success probability or failure probability.

Bernoulli experiment: Tossing a dice, A = apprerance of number "one" with P(A) = 1/6.
Bernoulli chain: tossing the dice for n times and observing the event A = appearance of number "one". The probability for event A is 1/6 in each toss.

Binomial Distribution
A Bernoulli chain of length n is given. For each of the n tosses, a certain event A can occur with the probability p (and the counter-event ¬A with the probability q = 1-p). One is interested in X = number of test tosses where A occurs. X can have any of the values k = 0,1,2,…,n. The probability that A occurs exactly k times is:

Binomialverteilung Formel

X is a binomially distributed random variable and the probability distribution of X is called a binomial distribution with the parameters n,p.

Properties of the Binomial Distribution

Expected value μ = E(X) = n*p

Variance σ² = Var(X) = n*p*q = n*p*(1-p)

Further information can be found in "Teschl Mathematik für Informatiker Volume 2, Edition 3, Chapter 28.2, Page 306-312".

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