Bernoulli process - Bernoulli-Kette
Bernoulli experiment
An experiment with exactly two outcomes (success or failure) is called a Bernoulli trial or Bernoulli-Experiment (in German).
Repeating a Bernoulli experiment n times leads to a random experiment with n steps, called a Bernoulli process or Bernoulli-Kette (in German) if following is true:
- The events
$A_1, A_2, ..., A_n$
are independent events. $P(A_1) = P(A_2) = ... = P(A_n) = p$
p is also the parameter of the Bernoulli process.
each path with k successes and (n-k) failures, in the Bernoulli experiment, has probability
$${P(“path”) = p^k \cdot (1 - p)^{n-k}}$$
binomial distribution - Binomialverteilung
The number of paths for each outcome with k successes depends on the number of trials n.
This is calculated using the binomial coefficient.
$${{}^{n}C_{r} =\frac{n!}{(n-r)! \cdot r!} = \binom n r}$$
The binomial coefficent $\binom{n}{k}$
states the number of ways of selecting k successes from n trials (see counting principles).
formula - Formel von Bernoulli
To find the probability of k successes in n trials following formula is used.
$${P(X=k) = \binom{n}{ k} \cdot p^k \cdot (1 - p)^{n-k}}$$
where p is the probability of success and (1 - p) is the probability of failure.
more notation
Another possible notation you may see is
$$X \sim B(n,p)$$
exercises
1
Es wird viermal gewürfelt. Berechne die Wahrscheinlichkeitsverteilung (probablity distribution) für S, “die Anzahl von gewürfelten Sechsern”.
hint: P(S = k)
2
Geburtsraten in Deutschland zeigen das 51% der Babys männlich sind. Berechne die Wahrscheinlichkeit, dass in einer Familie mit vier Kindern, mindestens drei Mädchen sind ("… a family of four children will contain three or more girls”).
binomial distribution table
$${B(n ; p ; k) = \binom{n}{ k} \cdot p^k \cdot (1 - p)^{n-k}}$$
How to use the table.
To find $P(X = 3)$
where, $n=4$
, $p=0.2$
and $k=3$
$B(4 ; 0,2 ; 3) = 0.0256$
Geogebra App
cumulative binomial distribution table - kumulierte Binomialverteilung F(n;p;k)
$${F(n ; p ; k) = B(n ; p ; 0) + B(n ; p ; 2) + … + B(n ; p ; k)}$$
For probabilities p greater than 0.5, the symmetry of the binomial distribution is used to find the cummulative binomial distribution.
Then you use the right hand side to find k, read off the probability using p below the table and subtract the value from 1.
TODO - examples with image of table and how to use the CASIO calculator.
Galton board
rosettacode - many different programs in various programming languages.