You will be an engineer in 100 days --Day 63 --Programming --Probability 1

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About probability

Looking up in the dictionary, it seems to be the ratio to all phenomena.

In programming, this probability calculation is also common.

Also, statistics as well as probability are one of the fields we deal with. If you go as a programmer, about terms and calculation methods I think it's better to keep it down.

Probability term

I picked up the terms of probability that I want to suppress.

the term	English	meaning
probability	Probability	Rate of events
permutation	Permutation	Arrange r pieces in a row in the order of taking out r pieces from n different pieces.
combination	Combination	Extracting r out of n different ones
Factorial	Factorial	The product of an integer from a positive integer to 1
Independence		The results of each other do not affect each other
Trial		The act of trying to find the probability
Repeated trial		An attempt to repeat the same action without changing the conditions
Expected value	Expected value	Mean of values obtained in one trial
Event	Event	What happens
Product event		Of the two events A and B, events where A and B occur at the same time
All events		A summary of all the consequences
Elementary event	Atomic event	Events that cannot be further subdivided
Complementary event		Events other than one
Mutual exclusivity		Events that do not occur at the same time
Empty event		An event that does not exist
Venn diagram		A visual diagram of the relationships between multiple sets and the extent of the sets
Odds	odds	Numerical value showing probability in probability theory
Probability distribution	probability distribution	Represents the probability of taking each value for a random variable
Random variable	random variable	The value assigned to what can happen
Discrete type(Random variable)	Discrete	とびとびの数になるRandom variable
Continuous type(Random variable)	Continuous	幾らでも細かく刻むことができるRandom variable

Probability calculation

Python has a library to help you do probability calculations You can calculate the probability relatively easily.

Factorial

The factorial can be calculated with factorial methods such as the math library.

Since the calculation method is the product of a certain positive integer to an integer from 1. If it is the factorial of 4, it becomes 4 * 3 * 2 * 1 (4!) And becomes 24.

import math

print(math.factorial(4))
print(math.factorial(10))

24 3628800

permutation

Permutation is the number when r are selected from n different ones and arranged in a row`.

Permutations can be generated and enumerated from lists (arrays). ʻIt is found by permutations in the itertools` library.

ʻItertools.permutations (list value, number to choose) `

In this case, the returned value itself is not a list If you want to see the values, you need to convert them to a list.

The following is the permutation when 2 out of 3 are selected and arranged.

import itertools

l = ['A', 'B', 'C']

p = itertools.permutations(l, 2)
print(list(p))

[('A', 'B'), ('A', 'C'), ('B', 'A'), ('B', 'C'), ('C', 'A'), ('C', 'B')]

Total number of permutations

If the number is small, it can be listed, but if the number of permutations is too large, it cannot be displayed. If you only want the total number, you can use math.factorial to calculate it.

The formula is n! / (N --r)!, So the total number of permutations to select 5 from 7 is

n , r = 7 , 5
math.factorial(n) // math.factorial(n - r)

2520

Now you can find it.

combination

The combination is the number when selecting r from different n, and does not consider the order like a permutation.

Permutations can be generated and enumerated from lists (arrays). ʻIt is found in combinations of the itertools` library.

ʻItertools.combinations (list value, number to choose) `

The enumeration of combinations to select 2 out of 5 is as follows.

l = ['a', 'b', 'c', 'd','e']
c = itertools.combinations(l, 2)
print(list(c))

[('a', 'b'), ('a', 'c'), ('a', 'd'), ('a', 'e'), ('b', 'c'), ('b', 'd'), ('b', 'e'), ('c', 'd'), ('c', 'e'), ('d', 'e')]

In this case, there is no duplication between elements.

To make a combination by taking into account the duplication of elements Use combinations_with_replacement.

ʻItertools.combinations_with_replacement (list value, number to choose) `

l = ['a', 'b', 'c']
c = itertools.combinations_with_replacement(l, 2)
print(list(c))

[('a', 'a'), ('a', 'b'), ('a', 'c'), ('b', 'b'), ('b', 'c'), ('c', 'c')]

Total number of combinations

If you only want the total number, you can use math.factorial to calculate it.

The formula is n! / (R! * (N --r)!), So the total number of combinations to select 3 from 7 is It will be as follows.

n , r = 7 , 3
math.factorial(n) // (math.factorial(n - r) * math.factorial(r))

Cartesian product (direct product of multiple lists)

To create a direct product of multiple lists, use product in the ʻitertools` library.

ʻItertools.product (list value, list value) `

l1 = ['a', 'b', 'c']
l2 = ['X', 'Y']
p = itertools.product(l1, l2)
print(list(p))

[('a', 'X'), ('a', 'Y'), ('b', 'X'), ('b', 'Y'), ('c', 'X'), ('c', 'Y')]

If you specify the number of repetitions in the argument repeat, the iterable object is used repeatedly. You can generate a direct product.

ʻItertools.product (list value, repeat = number of times) `

l1 = ['a', 'b', 'c']
p = itertools.product(l1, repeat=2)
print(list(p))

('a', 'a'), ('a', 'b'), ('a', 'c'), ('b', 'a'), ('b', 'b'), ('b', 'c'), ('c', 'a'), ('c', 'b'), ('c', 'c')]

Summary

I will touch on probability, but first let's learn terms. Tomorrow is about calculating expected value.

37 days until you become an engineer

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