(c) Albert Sorribas, Ester Vilaprino, Montse Rue, Rui Alves.
Biomodels Grup, Departament de Ciencies Mediques Basiques.
Universitat de Lleida, Institut de Recerca Biomedica de Lleida (IRBLleida

Convergence of the relative frequency to probability

Simulation of observing the relative frequency of an event of a given
probability in a series of observations.
The relative frequency tends to the probability for
a large number of observations

LEGEND

Blue line:

Prediction 95% interval where we expect to find
the relative frequency with the especified number of events.

Red line:

True probability. Samples are obtained ramdonly
as events of that probability.

Black dashed line:

Prediction 95% interval for the relative frequency
of the event at a given number of observations.

Empty dots:

Observed relative frequency at a given
number of observations.

(c) Albert Sorribas, Ester Vilaprino, Montse Rue, Rui Alves.
Biomodels Grup, Departament de Ciencies Mediques Basiques.
Universitat de Lleida, Institut de Recerca Biomedica de Lleida (IRBLleida

The binomial model

The binomial model allows for computing the probability of each of the possible results of observing how
many times an event will occur in a given sample. The model assumes that in each observation the
probability of the event is constant. The probability of observing \(x\) events
in \(n\) observations is computed as:
$$P(X=x)=
\begin{pmatrix}
n\\
x
\end{pmatrix} p^x (1-p)^{n-x}=\frac{n!}{x!(n-x)!}p^x (1-p)^{n-x}$$

Probability of a given result

Probability within a range

Expected results

In the left figure, we show the probabilities of all
the possible results in a sample of the indicated size. In the right figure, the
possible results are shown as relative frequencies. The table shows the numeric
values for the probabilities of the results. Only results
with a probability higher than 0.01 are shown

(c) Albert Sorribas, Ester Vilaprino, Montse Rue, Rui Alves.
Biomodels Grup, Departament de Ciencies Mediques Basiques.
Universitat de Lleida, Institut de Recerca Biomedica de Lleida (IRBLleida

Distribution of relative frequency

Perform a simulation on the relative fequency of an event of a given probability. The red
lines in the figure indicates the expected limits for the relative frequency for the
95% of the samples with n observations in each sample. You can indicate the value of
probability and the number of observations per sample. Then, you can especify a number
of samples and the scale of the figure. In the results panel, the percentage of
samples within the theoretical limits is shown. The proportion of samples within the
limits should by around a 95%. Try increasing the number of samples so that this
proportion converges towards 95%.

Expected 95% results

Observed results

(c) Albert Sorribas, Ester Vilaprino, Montse Rue, Rui Alves.
Biomodels Grup, Departament de Ciencies Mediques Basiques.
Universitat de Lleida, Institut de Recerca Biomedica de Lleida (IRBLleida

Probability of all the possible outcomes for a family with 4 descendants
where both parents are Aa

The result with one descendant AA, one aa, and two Aa has the maximum
probability, but not a high probability. In fact, it is much more probable
to observe any other result.

(c) Albert Sorribas, Ester Vilaprino, Montse Rue, Rui Alves.
Biomodels Grup, Departament de Ciencies Mediques Basiques.
Universitat de Lleida, Institut de Recerca Biomedica de Lleida (IRBLleida