Physical situation: Toss a coin. Randomness enters because we believe that the coinmay turn up head or tail and that it is inherently unpredictable.The corresponding probability model: Since there are two outcomes, the sample space Ω = {0, 1}(where we use 1 for heads and 2 for tails) is a clear choice. What about elementary probabilities?Under the equal chance hypothesis, we may take p0 = p1 =12. Then we have a probability modelfor the coin toss.If the coin was not fair, we would change the model by keeping Ω = {0, 1} as before but lettingp1 = p and p0 = 1 − p where the parameter p ∈ [0, 1] is fixed.Which model is correct? If the coin looks very symmetrical, then the two sides are equally likelyto turn up, so the first model where p1 = p0 =12is reasonable. However, if the coin looks irregular,then theoretical considerations are usually inadequate to arrive at the value of p. Experimentingwith the coin (by tossing it a large number of times) is the only way.There is always an approximation in going from the real-world to a mathematical model. Forexample, the model above ignores the possibility that the coin can land on its side. If the coin isvery thick, then it might be closer to a cylinder which can land in three ways and then we wouldhave to modify the model…

Basic theoretical probability