Probability

Probability

Listed below is all the content related to probability. 

Probability is the measure of the likelihood that an event will occur.[1] See glossary of probability and statistics. Probability is quantified as a number between 0 and 1, where, loosely speaking,[2] 0 indicates impossibility and 1 indicates certainty.[3][4] The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

Statistics and Probability

Exam Prep Study Notes (Things to remember) 

Mean
The mean of a set of numbers is the sum of the numbers in the set divided by the number of values in that set
Median
Given a set of numbers, the median is the middle number in that set when the numbers are arranged in an ascending (smallest to the largest) order. For a set with an even number of values, the average of the two middle numbers is taken
Mode
is the number that appears most frequently in a given set
.....
.......

 

Tips

Graphs

Use a bar graph to compare data

Use a line graph to show trends over time

Circle graph is used to show the parts of a whole. The percentages add up to 100%

Use a scatter plot to show trends when the data points are not connected

Impossible event
Probability of an impossible event is 0
Certain event
Probability of a certain event is 1
,,,,,
,,,,,,,

 

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Combination

Combination

In mathematics, a combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.

More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient

which can be written using factorials as whenever , and which is zero when k > 0. The set of all k-combinations of a set S is often denoted by .

Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection, k-multiset, or k-combination with repetition are often used. If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.

Although the set of three fruits was small enough to write a complete list of combinations, with large sets this becomes impractical. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.

Video: Combinations vs. Permutations

Number of k-combinations

3-element subsets of a 5-element set

The number of k-combinations from a given set S of n elements is often denoted in elementary combinatorics texts by  , or by a variation such as  ,  , , or even (the latter form was standard in French, Romanian, Russian, Chinese and Polish texts. The same number however occurs in many other mathematical contexts, where it is denoted by (often read as "n choose k"); notably it occurs as a coefficient in the binomial formula, hence its name binomial coefficient. One can define  for all natural numbers k at once by the relation

from which it is clear that

and further,

for k > n.

Video: Sum of binomial coefficients

To see that these coefficients count k-combinations from S, one can first consider a collection of n distinct variables Xs labeled by the elements s of S, and expand the product over all elements of S:

 

it has 2n distinct terms corresponding to all the subsets of S, each subset giving the product of the corresponding variables Xs. Now setting all of the Xs equal to the unlabeled variable X, so that the product becomes (1 + X)n, the term for each k-combination from S becomes Xk, so that the coefficient of that power in the result equals the number of such k-combinations.

Binomial coefficients can be computed explicitly in various ways. To get all of them for the expansions up to (1 + X)n, one can use (in addition to the basic cases already given) the recursion relation

for 0 < k < n, which follows from (1 + X)n = (1 + X)n − 1(1 + X); this leads to the construction of Pascal's triangle.

For determining an individual binomial coefficient, it is more practical to use the formula

 .

The numerator gives the number of k-permutations of n, i.e., of sequences of k distinct elements of S, while the denominator gives the number of such k-permutations that give the same k-combination when the order is ignored.

When k exceeds n/2, the above formula contains factors common to the numerator and the denominator, and canceling them out gives the relation

 

for 0 ≤ kn. This expresses a symmetry that is evident from the binomial formula, and can also be understood in terms of k-combinations by taking the complement of such a combination, which is an (nk)-combination.

Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember:

where n! denotes the factorial of n. It is obtained from the previous formula by multiplying denominator and numerator by (nk)!, so it is certainly inferior as a method of computation to that formula.

The last formula can be understood directly, by considering the n! permutations of all the elements of S. Each such permutation gives a k-combination by selecting its first k elements. There are many duplicate selections: any combined permutation of the first k elements among each other, and of the final (n − k) elements among each other produces the same combination; this explains the division in the formula.

Example of counting combinations

Video: Combinations Counting Example

As a specific example, one can compute the number of five-card hands possible from a standard fifty-two card deck as:

Alternatively one may use the formula in terms of factorials and cancel the factors in the numerator against parts of the factors in the denominator, after which only multiplication of the remaining  

Another alternative computation, equivalent to the first, is based on writing

 

which gives

 .

When evaluated in the following order, 52 ÷ 1 × 51 ÷ 2 × 50 ÷ 3 × 49 ÷ 4 × 48 ÷ 5, this can be computed using only integer arithmetic. The reason is that when each division occurs, the intermediate result that is produced is itself a binomial coefficient, so no remainders ever occur.

Using the symmetric formula in terms of factorials without performing simplifications gives a rather extensive calculation:

Enumerating k-combinations

One can enumerate all k-combinations of a given set S of n elements in some fixed order, which establishes a bijection from an interval of  integers with the set of those k-combinations. Assuming S is itself ordered, for instance S = { 1, 2, …, n }, there are two natural possibilities for ordering its k-combinations: by comparing their smallest elements first (as in the illustrations above) or by comparing their largest elements first. The latter option has the advantage that adding a new largest element to S will not change the initial part of the enumeration, but just add the new k-combinations of the larger set after the previous ones. Repeating this process, the enumeration can be extended indefinitely with k-combinations of ever larger sets. If moreover the intervals of the integers are taken to start at 0, then the k-combination at a given place i in the enumeration can be computed easily from i, and the bijection so obtained is known as the combinatorial number system. It is also known as "rank"/"ranking" and "unranking" in computational mathematics.[6][7]

There are many ways to enumerate k combinations. One way is to visit all the binary numbers less than 2n. Choose those numbers having k nonzero bits, although this is very inefficient even for small n (e.g. n = 20 would require visiting about one million numbers while the maximum number of allowed k combinations is about 186 thousand for k = 10). The positions of these 1 bits in such a number is a specific k-combination of the set { 1, …, n }.[8] Another simple, faster way is to track k index numbers of the elements selected, starting with {0 .. k−1} (zero-based) or {1 .. k} (one-based) as the first allowed k-combination and then repeatedly moving to the next allowed k-combination by incrementing the last index number if it is lower than n-1 (zero-based) or n (one-based) or the last index number x that is less than the index number following it minus one if such an index exists and resetting the index numbers after x to {x+1, x+2, …}.

Number of combinations with repetition

Video: Permutations and Combinations Problems - 3 Digit Even Number (Repetition Allowed)

A k-combination with repetitions, or k-multicombination, or multisubset of size k from a set S is given by a sequence of k not necessarily distinct elements of S, where order is not taken into account: two sequences define the same multiset if one can be obtained from the other by permuting the terms. In other words, the number of ways to sample k elements from a set of n elements allowing for duplicates (i.e., with replacement) but disregarding different orderings (e.g. {2,1,2} = {1,2,2}). Associate an index to each element of S and think of the elements of S as types of objects, then we can let denote the number of elements of type i in a multisubset. The number of multisubsets of size k is then the number of nonnegative integer solutions of the Diophantine equation:[9]

If S has n elements, the number of such k-multisubsets is denoted by,

a notation that is analogous to the binomial coefficient which counts k-subsets. This expression, n multichoose k, can also be given in terms of binomial coefficients:

This relationship can be easily proved using a representation known as stars and bars.

A solution of the above Diophantine equation can be represented by stars, a separator (a bar), then more stars, another separator, and so on. The total number of stars in this representation is k and the number of bars is n - 1 (since no separator is needed at the very end). Thus, a string of k + n - 1 symbols (stars and bars) corresponds to a solution if there are k stars in the string. Any solution can be represented by choosing k out of k + n - 1 positions to place stars and filling the remaining positions with bars. For example, the solution of the equation can be represented by

The number of such strings is the number of ways to place 10 stars in 13 positions, which is the number of 10-multisubsets of a set with 4 elements.

Bijection between 3-subsets of a 7-set (left) and 3-multisets with elements from a 5-set (right).
This illustrates that  .

As with binomial coefficients, there are several relationships between these multichoose expressions. For example, for ,

This identity follows from interchanging the stars and bars in the above representation.

 

Example of counting multisubsets

For example, if you have four types of donuts (n = 4) on a menu to choose from and you want three donuts (k = 3), the number of ways to choose the donuts with repetition can be calculated as

This result can be verified by listing all the 3-multisubsets of the set S = {1,2,3,4}. This is displayed in the following table. The second column shows the nonnegative integer solutions of the equation and the last column gives the stars and bars representation of the solutions.

No. 3-Multiset Eq. Solution Stars and Bars
1 {1,1,1} [3,0,0,0]  
2 {1,1,2} [2,1,0,0]  
3 {1,1,3} [2,0,1,0]  
4 {1,1,4} [2,0,0,1]  
5 {1,2,2} [1,2,0,0]  
6 {1,2,3} [1,1,1,0]  
7 {1,2,4} [1,1,0,1]  
8 {1,3,3} [1,0,2,0]
9 {1,3,4} [1,0,1,1]
10 {1,4,4} [1,0,0,2]  
11 {2,2,2} [0,3,0,0]  
12 {2,2,3} [0,2,1,0]  
13 {2,2,4} [0,2,0,1]  
14 {2,3,3} [0,1,2,0]  
15 {2,3,4} [0,1,1,1]  
16 {2,4,4} [0,1,0,2]  
17 {3,3,3} [0,0,3,0]  
18 {3,3,4} [0,0,2,1]  
19 {3,4,4} [0,0,1,2]  
20 {4,4,4} [0,0,0,3]  

 

 

 

 

 

 

Video: Combinations (as subsets)

Permutation

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Factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,

Video: Factorials ; Evaluating Expressions With Factorials

The value of 0! is 1, according to the convention for an empty product.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects).

n

n!

0 1
1 1
2 2
3 6
4 24
5 120
6 720
7 5040
8 40320
9 362880
10 3628800
11 39916800
12 479001600
13 6227020800
14 87178291200
15 1307674368000
16 20922789888000
17 355687428096000
18 ..............

..........

 


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Venn diagram

Venn diagram

Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves.

Video: What's a Venn Diagram, and What Does Intersection and Union Mean?

A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to easily read visualizations; for example, the set of all elements that are members of both sets S and T, S ∩ T, is represented visually by the area of overlap of the regions S and T. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics, and computer science.

A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.

Example

Sets A (creatures with two legs) and B (creatures that can fly)

This example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged. The blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. It is important to note that this overlapping region would only contain those elements (in this example creatures) that are members of both set A (two-legged creatures) and are also members of set B (flying creatures.)
Humans and penguins are bipedal, and so are then in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all be represented by points outside both circles.
The combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly (or both).
The region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles.

 

Video: Math Sets - Venn Diagram Part 1

 

Overview

Intersection of two sets
 

Union of two ....

 

 

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Probability

Probability

Quicklinks

What is probability?

What are the different types of probability?

Define sample space, power set and events

How do you assign probability to an event?

How to denote the probability of an event?

What is the opposite or complement of an event?

What is the joint probability of two events?

What is the probability of independent events?

What is the probability of both the events occurring if they are mutually exclusive?

What is the probability of either of the events occurring if they are mutually exclusive?

What is the probability for not mutually exclusive events?

What are the rules of probability?

Let's Review

Answer

Test Your Knowledge

What is probability?

Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. 
A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). 

Video: Probability for Beginners : Solving Math Problems

What are the different types of probability?

There are two types of probability, namely Theoretical and Experimental probability.
Theoretical probability = Number of favorable outcomes / Total number of outcomes
Experimental probability = Number of favorable outcomes / Total number of trials
Video: Types of probability

Define sample space, power set and events

Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a dice can produce six possible results. One collection of possible results gives an odd number on the dice. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the dice falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.

Video: Experiment, Outcome, Sample space, and event

How do you assign probability to an event?

A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.

Video: Simple probability

How to denote the probability of an event?

The probability of an event A is written as .......

 

 

Let's Review

  1. ____ is the measure of the likelihood that an event will occur.
  2. Probability is quantified as a number between ____ and ____.
  3. There are two types of probability, namely ____ and ____ probability.
  4. Theoretical probability = Number of favorable outcomes / Total number of ____.
  5. Experimental probability = Number of favorable outcomes / Total number of ____.
  6. The collection of all possible results is called the ____ ____ of the experiment. 
  7. The ____ ____ of the sample space is formed by considering all different collections of possible results.
  8. The probability of an event A is written as ____ , ____ , or ____ . 
  9. If two events A and B occur on a single performance of an experiment, this is called the ____ or ____ ____ of A and B, denoted as 
  10. If two events, A and B are independent then the ____ probability is for example, if two coins are flipped the chance of both being heads is.
  11. If two events are mutually exclusive then the probability of ____ occurring is denoted as 
  12. If two events are mutually exclusive then the probability of ____ occurring is denoted as .  For example, the chance of rolling a 1 or 2 on a six-sided die is 

    Answer

    1. Probability is the measure of the likelihood that an event will occur. 
    2. Probability is quantified as a number between 0 and 1.
    3. There are two types of probability, namely Theoretical and Experimental probability.
    4. Theoretical probability = Number of favorable outcomes / Total number of outcomes.
    5. Experimental probability = Number of favorable outcomes / Total number of trials.
    6. The collection of all possible results is called the sample space of the experiment. 
    7. The power set of the sample space is formed by considering all different collections of possible results.
    8. The probability of an event A is written as , or 
    9. If two events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B, denoted as 
    10. If two events, A and B are independent then the joint probability is for example, if two coins are flipped the chance of both being heads is.
    11. If two events are mutually exclusive then the probability of both occurring is denoted as .
    12. If two events are mutually exclusive then the probability of either occurring is denoted as .  For example, the chance of rolling a 1 or 2 on a six-sided die is  

    Test Your Knowledge

     


     

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