Introduction to basic probability.

Let us start with the notion of a sample space. A sample space describes the set of all possible outcomes when performing an experiment.

Let us consider a simple coin throw. The sample space is simply {H, T}. We can get either Heads or Tails as a result.

sample_space_coin = {"H", "T"}

Let’s consider the experiment that we throw a coin three times. The state space contains all possible combinations of Heads and Tails at each turn.

from itertools import product

sample_space_coins = set(product(sample_space_coin,repeat=3))
sample_space_coins

{('H', 'H', 'H'),
 ('H', 'H', 'T'),
 ('H', 'T', 'H'),
 ('H', 'T', 'T'),
 ('T', 'H', 'H'),
 ('T', 'H', 'T'),
 ('T', 'T', 'H'),
 ('T', 'T', 'T')}

An event is a possible outcome of our experiment. Let’s consider the event that Heads showed at least two times up.

event = {s for s in sample_space_coins if s.count("H") >= 2}
event

{('H', 'H', 'H'), ('H', 'H', 'T'), ('H', 'T', 'H'), ('T', 'H', 'H')}

The frequentist approach states the following: If all possible outcomes are equally likely an event E occurs with frequency #E/#S where #E is the number of elements in the Event and #S is the number of elements in the State Space. For our particular example we can do it very simply in Python.

from fractions import Fraction

Fraction(len(event)/len(sample_space_coins))

Similar we can consider a repeated dice throw. We identify each side of the dice with a number from 1-6. That means our sample space for a single dice throw is {1,...,6}.

sample_space_dice = {1,2,3,4,5,6}

sample_space_dices = set(product(sample_space_dice, repeat=2))
sample_space_dices

{(1, 1),
 (1, 2),
 (1, 3),
 (1, 4),
 (1, 5),
 (1, 6),
 (2, 1),
 (2, 2),
 (2, 3),
 (2, 4),
 (2, 5),
 (2, 6),
 (3, 1),
 (3, 2),
 (3, 3),
 (3, 4),
 (3, 5),
 (3, 6),
 (4, 1),
 (4, 2),
 (4, 3),
 (4, 4),
 (4, 5),
 (4, 6),
 (5, 1),
...
 (6, 2),
 (6, 3),
 (6, 4),
 (6, 5),
 (6, 6)}

From the above we see that there are 36 possibilities in total: 6 for each dice throw, Let’s consider the event that the sum of two dice throws is four or less.

event = {s for s in sample_space_dices if sum(s) <= 4}
event

{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)}

Fraction(len(event), len(sample_space_dices))

Fraction(1, 6)

We see that the probability of this event to occur is 1/6.

If you want to see more about the topics discussed here you may find the corresponding lecture by Harry Crane interesting.