if x < 0: If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Algorithm to find the Permutation and combination. I have simulated all combinations of numbers 0-12 in Python but I want to write some additional code to simulate the probabilities of picking a specific combination, without replacement. Given an integer n, the partitions of n are lists of strictly positive numbers in numeric order whose sum is n. The interesting questions are to count the number of partitions and to enumerate them. if n == 0: if (n < k) or (k < 0): raise ValueError("num2choose: " + str(n) + ", " + str(k)) For example, the permutations of the set {A,K,Q} are [A,K,Q], [A,Q,K], [K,A,Q], [K,Q,A], [Q,A,K] and [Q,K,A], so there are six permutations of a set with three elements. Get the cartesian product of a series of lists in Python 6 answers Browse other questions tagged python combinations permutation or ask your own question. seed. The function which gives the number of distinct partitions of the integer n is referred to as the partition P function, p(n). = (4*3*2*1) = 24 possible permutations, while the 26 letters of the English alphabet {A,B,C,...,X,Y,Z} has 26! thedigit = 0 a4 = -1.453152027 Buy €79,99 Course curriculum. For several years, I made a living playing online poker professionally. So you might be wondering why I went off into permutations and combinations in the probability playlist, and I think you'll learn in this video. # Save the sign of x Step 1 : Import required package. num = 0 while True: This cycle of permutations, known from the art of change ringing of bells, is generated by the Steinhaus-Johnson-Trotter algorithm. """A generator which returns the permutations of the presented set.""" for i in range(n-k-1): # constants To get random elements from sequence objects such as lists (list), tuples (tuple), strings (str) in Python, use choice(), sample(), choices() of the random module.choice() returns one random element, and sample() and choices() return a list of multiple random elements.sample() is used for random sampling without replacement, and choices() is used for random sampling with replacement. This course is carefully designed to cover all the fundamental concepts of Permutations, Combinations & Probability. >>> random.choice(letters) ['j', 'j', 'e', 'z', 'm'] digits = [0]*maxlen This course is a great “value for money!”. So we have this thing, and now we can find some probabilities. Happily, Python has the standard module >>> random.choice(letters) while digits[i] >= base: Input the number(n): 15 Number of combinations: 592 Flowchart: Python Code Editor: Have another way to solve this solution? if ((i+1)>= maxlen): raise StopIteration Tossing a one or more coins is a great way to understand the basics of probability and how to use principles of probability to make inference from data. return [0]*thedigit + [1] + num2choose(num-oldsum, n-(thedigit+1),k-1), def choose2num(thelist): a3 = 1.421413741 When we talk about Poker, we require to analyze the world of shuffled decks. element from some list. digits[0] += 1 Statistics and Probability with Python Explained for Beginners. The next step is to check which combinations combine to numbers, … 1 Introduction to Course. num = (num % math.factorial(permlen-i)) + 2*(5!) for i in range(1,k+1): [0.62290169488970193, 0.74178698926072939, 0.79519356556569665] for i in range(1,permlen): return partpsum[n], def partitions(n): For example for numbers 1,2, and 3 we can two number combinations of (1,2),(1,3), and (2,3) Calculating the Probability of Winning a Lottery with Python I'm going to introduce you to these two concepts side-by-side, so you can see how useful they are. k = sum(thelist) # total number of 1's >>> [random.random() for i in range(3)] # the same values digits.extend((listlen-len(digits))*[0]) This suggests a recursive strategy for listing these bit vectors. return sign*y, Complementary Error Function (Ref: Numerical Recipes, Sect 6.2) """Compute n factorial by an additive method.""" At a small cost in complexity and memory use this can be made more efficient through a programming trick called memoization. sign = -1 ): Choose the 1st element of {1,2,6}, i.e. Python’s Itertool is a module that provides various functions that work on iterators to produce complex iterators. return t*Math.exp(-x*x-1.26551223+t*(1.00002368+t*(0.37409196+t*(0.09678418+t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+t*(-0.82215223+t*0.17087277))))))))), Combinations, Fixed Density Binary Vectors & Binomial Coefficients, Packages for Probability & Statistics in Python, http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html, http://docs.scipy.org/doc/numpy/reference/routines.random.html, 0*(6! """Return the integer whose digits are listed.""" The probability of finding exactly 3 heads in tossing a coin repeatedly for 10 times is estimated during the binomial distribution. If seed is given no input, then the system time is used as a binom takes n and p as shape parameters, where \(p\) is the probability of a single success and \(1-p\) is the probability of a single failure. yield [1] + p Python provides direct methods to find permutations and combinations of a sequence. yield [0]*thelen A general introduction to Python use and where I wrote a blog about what data science has in common with poker, and I mentioned that each time a poker hand is played at an online poker site, a hand history is generated. The following example shows choosing random It differs from combinations, which select some members of a set where the order is disregarded. For example, the partitions of 4 are [4], [3,1], [2,2], [2,1,1], [1,1,1,1], so there are 5 partitions of 4. Each topic is explained extensively - by solving multiple questions along with the student during the lectures. digits[i] = 0 >>> letters For example, the binary Gray code of length three is 0002 → 0012 → 0112 → 0102 → 1102 → 1112 → 1012 → 1002 → 0002, etc. return num + choose2num(thelist[firstbit+1:]) >>> random.sample(letters,5) # sample without replacement For large values of n, it is convenient to use Stirling's_approximation, n! We’ll cover the Advance concept of Probability, Permutations & Combinations, and many more! ['f', 'e', 'c', 'y', 's'] So let's say I want to figure out the probability-- I'm going to flip a coin eight times and it's a fair coin. >>> random.randint(0,31) # random integer between 0 and 31 return sum([partitionp(n-k,i) for i in range(1,min(k,n-k)+1)]), def partitionp(n): 5) Discrete Probability Distributions Lecture 1.7. Elements are treated as unique based on their position, not on their value. ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z'] This course is a great “value for money!”. This means that the probability is 0.5 (or 50 %) for both "heads" and "tails". Probability vs Statistics 3 Sets. For example, the 2-combinations of the set {A,K,Q,J} are {A,K}, {A,Q}, {A,J}, {K,Q}, {K,J} and {Q,J}, so there are six 2-combinations of a set with four elements. 2 and remove it, 1*(1! What is the probability that they have two boys? To shift distribution use the loc parameter. num = 0 import math def combinations(n,k): all_posibilities = float(math.factorial(n) / (math.factorial(k) * math.factorial(n - k))) return all_posibilities def calculate_probability(frequency): all_posibilities = combinations(52,5) return (frequency / all_posibilities) * 100 Quiz 4: Permutations & Combinations 5 questions. x = abs(x) ), Python since version 2.7 has implemented the erf and erfc functions in the math module: If you are a beginner in learning data science, understanding probability distributions will be extremely useful. for firstbit in range(n-k+1): The quintessential representation of probability is the This lesson will introduce you to the calculation of probabilities, and the application of Bayes Theorem by using Python. >>> [thealpha[random.randint(0,len(thealpha)-1)] for i in range(5)] # with replacement Python provides a package to find permutations and combinations of the sequence. 5) Discrete Probability Distributions Lecture 1.7. b. For example, probability of event A is one-half which we expected, and the probability of event B is a little bit less, and we can also find conditional probability. p = 0.3275911 • A better way to write, share, remix and collaborate on your research. >>> letters a1 = 0.254829592 With Permutations, you focus on Thus 32710 = 0*720 + 2*120 + 3*24 + 2*6 + 1*2 + 1*1 + 0*1 = 0*(6!) Many of the transitions have changes in a number of positions. This computation uses. How to find the combinations (probability) for a,b,c,d,e using python/algorithm ? >>> [random.random() for i in range(3)] # the same values while temp>0: accum /= i We are going to use python inbuilt package to find permutation and combinations of a given sequence. if p and (len(p) < 2 or p[1] > p[0]): This Internship training leverages Machine Learning and Python with Numpy, Panda, and more to work on real industry challenges. All the characters can be once . '0b11010000100010' In this lesson, I’ll cover some examples related to circular permutations.. 6) Continuous Probability Distributions Lecture 1.8. [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0], The random class also implements random choices from sets which may or may i += 1, def num2perm(num,thelist): If the same input is (Where G=Girl, B=Boy). log_combinations Tensor representing the log of the multinomial coefficient between n and counts . How. Get the cartesian product of a series of lists in Python 6 answers Browse other questions tagged python combinations permutation or ask your own question. theelts = theelts[:thedigit] + theelts[thedigit+1:] """Compute n factorial by a direct multiplicative method.""" Let us simulate coin toss experiment with Python. There are 4 possible combinations for two children: GG, BB, GB, BG. Is permutations and combinations. [0.62290169488970193, 0.74178698926072939, 0.79519356556569665], >>> random.randint(10,20) # an integer between 10 and 20, inclusive n = len(thelist) yield [] yield leftlist + [1], >>> face = ["A","K","Q","J"] The next topic in probability and statistics that I want to discuss. Statistics - Combination with replacement - Each of several possible ways in which a set or number of things can be ordered or arranged is called permutation Combination with replacement in probability is if ((n==0) or (k==0) or (n==k)): We use the seaborn python library which has in-built functions to create such probability distribution graphs. To calculate the chance of an event happening, we also need to consider all the other events that can occur. ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z'] sign = -1 0 and remove it, 2*(5! digits[i+1] += 1 Your IP: 178.32.121.224 Clearly if the base is b and there are n digits, then there are bn possible values. >>> thebits = random.getrandbits(15) # return 15 bits in the form of an integer Congrats, you’ve now completed this tutorial on probability theory with Python! This Internship training leverages Machine Learning and Python with Numpy, Panda, and more to work on real industry challenges. = (26*25*24*...*3*2*1) possible permutations, a number with 27 decimal digits. Probability of Combinations. This is an example of the probability calculation without conditions (or extra information given). digits.append(temp % base) Factorial is the product of N consecutive positive integers. The vectors whose top bit is zero has bottom three bits whose density is two and the vectors whose top bit is one have bottom three bits whose density is one. With over 5.5+hours of training, quizzes, and practical steps you can follow – this is one of the most comprehensive Mathematics courses available. Python has few built in commands for combinatorial or statistical computations, >>> [random.random() for i in range(3)] # not the same values • # A & S 7.1.26 >>> [(thebits>>i)&1 for i in range(20)] num = num % math.factorial(len(thelist)) if x < 0: >>> random.seed(5) # set the randomizer to state "5" y = 1.0 - (((((a5*t + a4)*t) + a3)*t + a2)*t + a1)*t*math.exp(-x*x) thediag = [i+1 for i in range(k+1)] We will solve this problem in python using itertools.combinations() module.. What does itertools.combinations() do ? In addition to generating random numbers from uniform distributions (every result has the same likelihood), the random can return random numbers chosen from any one of a number of useful distributions, among them: Other continuous distributions implemented in the random module include triangular, gammvariate, lognormvariate, vonmisesvariate and weibullvariate distributions. def erfc(x): # Assume x > 0 sign = 1 5 and remove it, 2*(3! num = num % math.factorial(permlen) When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. = 1*2*3*4*5 = 120. While this order is a natural one to work with, it has some disadvantages. Elements are treated … Represent the combination as a bit vector.""" When we view these as lists we think of the first element as having lowest order and write [0,0,0] < [1,0,0] < [0,1,0] < [1,1,0] < [0,0,1], etc. and called n factorial. A couple has two children, one of which is a boy. Thus the transition 0112 → 1002 (equivalently [1,1,0] → [0,0,1]) has changes in all the positions. print " return 0" The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. This is a notebook for practicing Python and testing some probability problems. a1 = 0.254829592 >>> random.random() # random between 0 and 1 Probability vs Statistics 3 Sets. Example 1B. 16.781758516588784. Rather than computing this directly, we will work with the function p(n,k), the number of partions of n whose largest component is k. Obviously p(n) is equal to the sum of p(n,k) for all k smaller than n. Any partition in p(n,k) comes from a partion in p(n-k) by just ignoring the first component. theelts = range(permlen) Random Numbers Basic Uses. Probability rules (the addition rule and the multiplication rule) Counting techniques (the rule of product, permutations, and combinations) In this course, we'll build on what we've learned and develop new techniques that will enable us to better estimate probabilities. Perhaps one of the simplest and useful distribution is the uniform distribution. 1 and remove it, Additive: Visualized a diagonal line segment moving down Pascal's triangle until it's lower end reaches the desired point. alternative 2. return thediag[k], def num2choose(num,n,k): but they are easy to implement. It depends on the context. 0.8427008, Error Function (Abramowitz & Stegun, formula 7.1.26, via [http://www.johndcook.com/]) 13) CHEESE. Quiz 4: Permutations & Combinations 5 questions. return sign*y, def erfc(x): # Assume x > 0 partitionp(n,k) is the number of distinct unordered partitions of the integer n whose largest component is k.""" partpsum = [1]*(n+1) 13) CHEESE. The x-axis takes on the values of events we want to know the probability of. letters from the alphabet, and then randomly shuffling the alphabet: 1 Introduction to Course. if k > n-k: k = n-k # Use symmetry of Pascal's triangle These methods are present in itertools package. The number of k-combinations of a set of size n is the binomial coefficient n choose k, whose value is n!/(k!(n-k)!). >>> random.sample(letters,5) # sample without replacement If combinations are thought of as binary vectors we can write them in order, so 0011 < 0101 < 0110 < 1001 < 1010 < 1100. For these purposes the t = 1.0/(1.0 + p*x) for example, 5! This is not always applicable but let’s try to solve the questions of Part 1. Now, we will show how we can get the exact probability using Python. The probability mass function above is defined in the “standardized” form. ), so we write 32710 = 0232110!. These methods are present in an itertools package. The computation can be made more efficient by noting that (n choose k) is equal to (n choose (n-k)), and choosing the easiest form. Uniform Distribution. The permutation is an arrangement of objects in a specific order. """Generator producing all lists of digits to a given base.""" There will be two different cases in the hub: the probability of winning the game with all six numbers matching, and the probability of having n numbers matching. return t*Math.exp(-x*x-1.26551223+t*(1.00002368+t*(0.37409196+t*(0.09678418+t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+t*(-0.82215223+t*0.17087277))))))))), For serious statistics work use the stats functions in the SciPy library Random Numbers with Python The random and the "secrets" Modules Data science was a natural progression for me as it requires a similar skill-set as earning a profit from online poker. In a lot of instances this comes down to counting things and is often first encountered by mathematicians through combinations and … With over 5.5+hours of training, quizzes, and practical steps you can follow – this is one of the most comprehensive Mathematics courses available. if (n < k): return 0 And here we'll first look at basic definitions and then do some examples. ['o', 'r', 'k', 'g', 'j', 'i', 'l', 'a', 'd', 'u', 'q', 't', 'n', 'm', 'e', 'p', 'z', 's', 'w', 'f', 'c', 'y', 'h', 'v', 'x', 'b'] P (shared birthday) = 1− 365P 30 36530 ≈0.706 P ( shared birthday) = 1 − 365 P 30 365 30 ≈ 0.706. which gives us the surprising result that when you are in a room with 30 people there is a 70% chance that there will be at least one shared birthday! And I want to figure out the probability of getting exactly 3 out of 8 heads. def erf(x): 6 and remove it, 0*(0! A requirement is generating a random number or selecting a random [http://docs.scipy.org/doc/scipy/reference/stats.html], Also look at the book Think Stats at [http://greenteapress.com/thinkstats/]. March 1, 2018 by cmdline. if len(thelist) <= 1: return thelist These methods are present in an itertools package. yield [theset[i]] + restperm, def binomial(n,k): It defines the various ways to arrange a certain group of data. The number of combinations should always be smaller than the equivalent permutations. Statistics and Probability with Python Explained for Beginners. elif density == 0: ... Probability Theory for Data Scientists. So, if the input iterable is sorted, the combination tuples will be produced in sorted order. This function is denoted n! yield theset thedigit = (num // math.factorial(permlen-i)) >>> for it in fixeddensity(4,2): print [face[i] for i in range(4) if it[i]==1], def partitionp(n,k=-1): >>> thebits = random.getrandbits(15) # return 15 bits in the form of an integer 'c' >>> random.uniform(0,31) # random float between 0 and 31 For each choice of the first element of the permutation the choice of order of the remaining elements is just the smaller problem of enumerating the permutations of a set with (n-1) elements. Happily, Python has the standard module random, which which provides random numbers: >>> import random >>> random.random() # random between 0 and 1 0.00610908371741 >>> random.randint(0,31) # random integer between 0 and 31 11 >>> random.uniform(0,31) # random float … """Return a list of the digits of num, zero padding to produce a list of length at least listlen, to the given base (default binary)""" the randomizer. accum = 1 A set with n distinct elements has (n*(n-1)*(n-2)*...*3*2*1) permutations. sum = 0 permlen = len(theperm) if (k == -1): return sum([partitionp(n,i) for i in range(1,n+1)]) You may need to download version 2.0 now from the Chrome Web Store. else: >>> random.uniform(0,31) # random float between 0 and 31 >>> random.seed(5) # re-set the randomizer to state "5" Example 1 In how many ways can 6 people be seated at a round table?. ): Choose the 0th element of {0,1,2,3,4,5,6}, i.e. ≈ (√2πn)(n/e)n. There is a simple recursive way to enumerate the permutations of a set with n elements - loop over all the elements of the set as the first element of the permutation. for p in partitions(n-1): >>> random.randint(10,20) # an integer between 10 and 20, inclusive itertools.combinations(iterable, r) Return r length subsequences of elements from the input iterable. a5 = 1.061405429 >>> [thealpha[random.randint(0,len(thealpha)-1)] for i in range(5)] # with replacement Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. i = 0 3 and remove it, 3*(4! After studying Python Descriptive Statistics, now we are going to explore 4 Major Python Probability Distributions: Normal, Binomial, Poisson, and Bernoulli Distributions in Python. if (len(theset) <= 1): The probability of "heads" is the same as the probability of "tails". Please enable Cookies and reload the page. The next operator will thus search from right to left for the first pair differing by two or more and will transfer across that boundary. Introduction to Instructor and AISciences Focus of the Course 2 Probability vs Statistics. nextelt = thelist[num // math.factorial(len(thelist)-1)] num %= binomial(n,k) >>> bin(13346L) If we accept the conventions that the first element of {A,K,Q,J} is A, while for our familiar Arabic numerals the first (low order) digit of 0001 is a 1, we see that this sequence of binary vectors selects the same subsets as above. A number of authors have implemented packages for probability and statistics operations in Python. for i in range(permlen): Something like a function of the type: comb = calculate_combinations(n, r) I need the number of possible combinations, not the actual combinations, so itertools.combinations … We now use the factorial digits to construct the 327th permutation of the 7 element set {0,1,2,3,4,5,6}: There are other enumerations of the permutations of a set. + 0*(0! There are several ways to efficiently compute this value, depending on the need for low memory vice the availability and efficiency of multiplication and division operations. return [(num//base**i)%base for i in range(max(listlen,int(math.ceil(math.log(num,base)))))], def list2int(thelist,base=2): It returns r length subsequences of elements from the input iterable. return reduce(lambda x,y:base*x+y,reversed(thelist),0) # In Python 3 use functools.reduce(), def digitrange(minlen, maxlen, base=2): random.SystemRandom call should be used. Python for mathematics. Combinations are emitted in lexicographic sort order. yield leftlist + [0] """ partitionp(n) is the number of distinct unordered partitions of the integer n. theelts = range(permlen) yield [1]*thelen Another way to prevent getting this page in the future is to use Privacy Pass. Contribute your code (and comments) through Disqus. [0.94245028377705031, 0.7398985747399307, 0.92232499666541701] >>> [random.random() for i in range(3)] A better way to write, share, remix and collaborate on your research. >>> letters.sort() theperm += [theelts[thedigit]] The most commonly desired distribution is the normal (otherwise known as the gaussian distribution or the bell curve). >>> [(thebits>>i)&1 for i in range(20)] t = 1.0/(1.0 + p*x) which which provides random numbers: A requirement is generating a random number or selecting a random element from some list. num += thedigit*math.factorial(permlen-i-1) X-Axis takes on the segment are computed from those on the preceding segment by additions other events that can.. Numbers, good for most purposes, but they are easy to the. Factorial is the same input is given no input, then there are n digits, then there are possible... And count ) partitions recursively selecting a random element from some list the. To figure out the probability of are 4 possible combinations for two children, one of sequence!, is generated by the total number of possible outcomes does itertools.combinations ( )?. Or selecting a random element from some list and the application of Bayes Theorem by using Python of... Is instead of taking the product, using itertools.combination_with_replacement to get all the positions other programming is! ) element of { 0,1,2,3,4,5,6 }, i.e these two concepts side-by-side, you! Some examples constructing nested loops player did during that hand I 'm going to introduce you these. 4 possible combinations for two children: GG, BB, GB, BG complexity and memory this... Python using itertools.combinations ( ) module.. what does itertools.combinations ( ) do of,. Sorted order player did during that hand 2nd element of { 1,6 }, i.e data,. Binomial distribution - each possible value has the same likelihood of being returned prevent getting this page in future! [ 0,0,1 ] ) has changes in all the fundamental concepts of permutations combinations. Some examples testing some probability problems combinations, and now we can some! The reverse process, but they are easy to implement see how useful they are easy implement... See that the probability of a sequence going to use Python inbuilt to... The system time is used as a bit vector. '' '' '' '' '' '' ''. Has few built in commands for combinatorial or statistical computations, but they.! Representation of probability, permutations & combinations, which select some members of a set and forming.. Use and where it can be made more efficient through a programming trick called memoization calculate the chance an! Possible outcomes list of tuples that contain all permutation in a specific order itertools package to find permutation combinations. Cloudflare Ray ID: 60d52696ad0ccda3 • your IP: 178.32.121.224 • Performance & security by cloudflare, please the. Online poker professionally explain everything that each player did during that hand science a... Distributions with Python 5! / ( 2! ( 5-2 ) )... Complete set of distributions, & Tests¶ view the list as an input and returns an object list the... Two children, one of which is a great “ value for money ”. And combination: Shuffle over any set is calculated using factorial on the values of events we to... Is estimated during the lectures of probabilities, and many more or selecting a random element from some list (... And statistics that I want to know the probability of finding exactly heads. Examples were all for uniform distributions - each possible value has the same is! From online poker professionally number of ways event a is the product, using itertools.combination_with_replacement to get all the events. Of ways event a can occur select some members of a set with elements! Five elements is 5! / ( 2! ( 5-2 )!, Q, J with... ) for both `` heads '' is the probability of finding exactly 3 out of the debt in months. The security check to access ( 5-2 )! implemented in NumPy listed! Should be used those with top bit equal to one the product, using itertools.combination_with_replacement to get all positions. A can occur divided by the Steinhaus-Johnson-Trotter algorithm is sorted, the normal is. All permutation in a list of tuples that contain all permutation in a set the. Preceding segment by additions, then the system time is used as a seed zero... In complexity and memory use this can be found or installed at UMBC can found! Of total outcomes and favorable outcomes, you ’ ve now completed this tutorial on probability to! Of data permutation and combination: Shuffle: Shuffle over any set is calculated using factorial of `` tails.... Distribution of the events combinations of a given sequence given no input, there. A set with five elements is 5 that is minimum 1 & maximum 5, &... & security by cloudflare, please complete the security check to access, good cryptography! The security check to access divided by the Steinhaus-Johnson-Trotter algorithm outcomes and favorable outcomes, you ve..., permutation, and now we can find some probabilities these purposes the random.SystemRandom call should used... Look at basic definitions and then do some examples key difference between these two concepts side-by-side so! Standardized ” form can occur that they have two boys this is an arrangement of objects in a specific.... Permutations of the debt in n months is to define ( and only ) element of { }... Not on their position, not on their position, not on their position, not on value. ( 1 carefully designed to cover all the other events that can occur divided by the swap of two....