A Noob’s perspective — Learning Data Science, ML and AI from scratch. Pt 6
Finite Sample spaces
Suppose S is finite, and S = {s1,s2,s4,s4, …, sn-1,sn}. Finite sample spaces allow us to calculate the probabilities of certain events in an efficient manner.
Now let’s say that A is a subset of S, then the probability P(A) is the sum of all the probabilities given that they belong in A
Example:
say that you are presented 4 toy balls, 2 red 1 green, 1 blue. What are the odds that you grab either a blue or red one.
P(red) = 1/2
P(blue) = P(green) = 1/4
P(A) = P(red) + P(blue) = 1/2 + 1/4 = 3/4
Addition rule
Say that we have a choice A that we could do in n-th ways, we’ll call it na ways, and We have another choice B with nb ways too. In the end we have (na + nb) ways of doing things.
For example.
imagine we could choose between having a burger or pizza; and we have the option between bacon and no bacon for the burger, and vegetarian, pineapple, or sausage pizza. We have in total 5 options. 2 for the burger and 3 for the pizza.
Replacement and order
imagine we pick 2 cards from a deck, without replacement, how many ways can we do this.
We care about order meaning (6♣, Q♦) not equal to (Q♦,6♣)
Answer
There’s 52 cards, pick one we are left with 51 cards. 52 . 51 = 2652. If that isn’t clear try to experiment with a smaller deck of cards on your own.
Counting techniques
Permutations.- An arrangement of n elements in a definite order is a permutation of the n elements.
Example
- how many ways can you arrange 246
- 6 ways. 246, 264, 426, 462, 624, 642
The general form can be proven with an example.
how many ways can i arrange 1,2,3 …, n-1, n.
look at it like the previous card example where we pick two. Now let’s imagine if we keep picking cards.
52 x 51 x 50 x 49 x … = 52!
Formal definition. The number of r-arrays we can make from a bigger set of n elements, where we use each element at most once, is called the number of permutations of n things taken r-at-a-time.
Example: how many ways can you take two vowels from a,e,i,o,u.
(5!)/(5–2)! = 5!/3! = 20
ae, ai, ao, au, ea, ei, eo, eu, ia, ie, io, iu, oa, oe, oi, ou, ua, ue, ui, uo
Combinations
Here we are interested in making only subsets of length r out n without any order.
Example:
- how many subsets can me make out of {1,2,3}
- {1,2},{,1,3}.{2,3}
