## Random Experiments |

Simulation of the dice experiment

Simulation of Buffon's coin experiment

Probability theory is based on the paradigm of a *random
experiment*; that is, an experiment whose outcome cannot be
predicted with certainty, before the experiment is run. We
usually assume that the experiment can be repeated indefinitely
under essentially the same conditions. This assumption is
important because probability theory is concerned with the
long-term behavior as the experiment is replicated. Naturally, a
complete definition of a random experiment requires a careful
definition of precisely what information about the experiment is
being recorded, that is, a careful definition of what constitutes
an *outcome.*

The *dice experiment* consists of rolling a pair of
(distinct) dice and recording the number spots showing on each
die.

*Buffon's coin experiment* consists of tossing a coin
with radius *r* < 1 on a floor covered with square
tiles of side length 1. The coordinates of the center of the coin
are recorded, relative to axes through the center of the square,
parallel to the sides.

** 1**. Run the simulation of the
dice experiment 100 times and observe the results.

** 2**. Run the simulation of
Buffon's coin experiment 100 times and observe the results.

Suppose that we have *n* experiments *E*_{1},
*E*_{2}, ..., *E*_{n}. We
can form a new, compound experiment by performing the *n*
experiments in sequence (*E*_{1} first, and then *E*_{2},
and so on). In particular, suppose that we have a basic
experiment. A fixed number (or even an infinite number) of
replications of the basic experiment is a new, compound
experiment. Many experiments turn out to be compound experiments
and moreover, as noted above, probability theory itself is based
on the idea of replicating an experiment.

** 3**. Identify each of the
following experiments as a compound experiment, based on
replicating a simpler experiment.

- The dice experiment
- Buffon’s coin experiment

The term parameter refers to a non-random quantity in a model that, once chosen, remains constant. Many probability models of random experiments have one or more parameters that can be adjusted to fit the physical experiment being modeled.

** 4**. Identify the parameter in the
Buffon's coin experiment.

## Probability Spaces |