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

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**.

__Example__

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:

*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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