### 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.

#### Examples and Simulations

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.

#### Compound experiments

Suppose that we have n experiments E1, E2, ..., En. We can form a new, compound experiment by performing the n experiments in sequence (E1 first, and then E2, 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.

1. The dice experiment
2. Buffon’s coin experiment

#### Parameters

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.