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HP 49g Manual: Understanding Probability Distributions with Factorials and Combinations

Models: 49g

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Page 153

Chapter 15

Probability Distributions

In this Chapter we provide examples of applications of the pre-defined probability distributions in the calculator.

The MTH/PROBABILITY.. sub-menu - part 1

The MTH/PROBABILITY.. sub-menu is accessible through the keystroke sequence „´. With system flag 117 set to CHOOSE boxes, the following functions are available in the PROBABILITY.. menu:

In this section we discuss functions COMB, PERM, ! (factorial), and RAND.

Factorials, combinations, and permutations

The factorial of an integer n is defined as: n! = n⋅ (n-1) ⋅ (n-2)…3⋅2⋅1. By definition, 0! = 1.

Factorials are used in the calculation of the number of permutations and combinations of objects. For example, the number of permutations of r objects from a set of n distinct objects is

n Pr = n(n − 1)(n − 1)...(n − r + 1) = n! /(n − r)!

Also, the number of combinations of n objects taken r at a time is

		 =	(	− 1)(		− 2)...( −		+ 1) =		!
	n		n n		n		n	r		n
		
						r!				r!(n − r)!
 r						r!				r!(n − r)!

Page 15-1

Image 153

HP 49g Chapter Probability Distributions, MTH/PROBABILITY.. sub-menu part, Factorials, combinations, and permutations

Contents

User’s manual Hp 49g+ graphing calculatorPrinting History Preface Table of Contents Calculations with real numbers Algebraic and arithmetic operations Calculations with complex numbersOperations with lists Solution to equationsMatrices and linear algebra VectorsCalculus Applications GraphicsMulti-variate Calculus Applications Vector Analysis ApplicationsDifferential Equations Probability DistributionsStatistical Applications Using SD cards Limited Warranty W-1Numbers in Different Bases Chapter Getting started Basic OperationsBatteries Contents of the calculator’s display Tool menu MenusIntroducing the calculator’s keyboard Setting time and datePage Selecting calculator modes Operating Mode √ 3.*5.-1/3.*3./23.3+EXP2.5 Lets try now the expression proposed earlier Number Format and decimal dot or comma Page Page Page Angle Measure Coordinate System Selecting CAS settings Selecting Display modes Explanation of CAS settingsSelecting the display font Selecting properties of the line editor Selecting properties of the Stack Selecting properties of the equation writer EQW ReferencesCalculator objects Chapter Introducing the calculatorEditing expressions in the stack Creating arithmetic expressionsPage Page 2L 1 + + 2 L + y Creating algebraic expressionsUsing the Equation Writer EQW to create expressions Page + 2 ⋅ 5 + + e  x + 2µ ⋅ ∆y Organizing data in the calculator Home directorySubdirectories Typing variable names VariablesCreating variables Page Page Checking variables contents Page Deleting variables Using function Purge in the stack in Algebraic modeUsing function Purge in the stack in RPN mode Undo and CMD functionsChoose boxes vs. Soft Menu Show Prog menu list and select Memory Page References Examples of real number calculations Chapter Calculations with real numbersPage Page Using powers of 10 in entering data Page Hyperbolic functions and their inverses Real number functions in the MTH menuPage Units menu Operations with unitsPage Available units Attaching units to numbers Operations with units Page Unit conversions Physical constants in the calculatorEngl Defining and using functions ‘LNx+1 + EXPx ‘ Reference Setting the calculator to Complex mode Chapter Calculations with complex numbersDefinitions Polar representation of a complex number Entering complex numbersSimple operations with complex numbers Cmplx menu through the MTH menu Cmplx menusCmplx menu in keyboard Function Droite equation of a straight line Functions applied to complex numbersReference Entering algebraic objects Chapter Algebraic and arithmetic operationsSimple operations with algebraic objects `‚¹ Collect Functions in the ALG menuPage Expansion and factoring using log-exp functions Expansion and factoring using trigonometric functionsOperations with transcendental functions Functions in the Arithmetic menu Horner function PolynomialsPcoef function VariableProot function Quot and Remainder functionsFractions Peval functionSIMP2 function Propfrac function Partfrac functionFcoef function Step-by-step operations with polynomials and fractions Froots function− 5X 2 + 3X − Reference Chapter Solution to equations Symbolic solution of algebraic equationsFunction Isol Function Solve Page Function Zeros Function SolvevxNumerical solver menu @SOLVE@ Polynomial EquationsSolve for coefficients ˜@SYMB@ Solving equations with one unknown through NUM.SLV Financial calculationsSolution to simultaneous equations with Mslv Page Reference Creating and storing lists Chapter Operations with listsOperations with lists of numbers Changing signAddition, subtraction, multiplication, division Functions applied to lists Lists of complex numbers Lists of algebraic objectsMTH/LIST menu ∆LIST MAP function SEQ functionChapter Vectors Entering vectorsTyping vectors in the stack Storing vectors into variables in the stack Using the Matrix Writer Mtrw to enter vectorsPage Page Addition, subtraction Simple operations with vectorsAbsolute value function Multiplication by a scalar, and division by a scalarMTH/VECTOR menu MagnitudeDot product Cross product Using the Matrix Writer Chapter Matrices and linear algebraEntering matrices in the stack Typing in the matrix directly into the stack Addition and subtraction Operations with matricesMultiplication Cij = ∑aik ⋅ bkj , for i = 1,2,K, m j = 1,2,K, n Page Characterizing a matrix The matrix Norm menu Solution of linear systemsFunction DET Function TraceUsing the numerical solver for linear systems =  a M x2 Solution by division of matrices Solution with the inverse matrixGraphs options in the calculator Chapter Graphics= 1 exp− x Plotting an expression of the form y = fxGenerating a table of values for a function Page Fast 3D plots Left-1 Right1 Near-1 Far Low High Step Indep Depnd Page Reference Calc Calculus menu Chapter Calculus ApplicationsLimits and derivatives Function limFunctions Deriv and Dervx Anti-derivatives and integrals Xdx = F x + CFunctions INT, INTVX, RISCH, Sigma and Sigmavx Definite integrals Infinite seriesFunctions TAYLR, TAYLR0, and Series Reference Partial derivatives Chapter Multi-variate Calculus ApplicationsΦ x, ydydx Multiple integralsDel operator Chapter Vector Analysis ApplicationsGradient ∇ = i ⋅ ∂ + j ⋅ ∂ + k ⋅ ∂ ∂x ∂y ∂zCurl DivergenceSolution to linear and non-linear equations Chapter Differential EquationsCALC/DIFF menu Function Ldec Yx = 5 ⋅ exp−x3 / 3 ⋅ expx3 / 3 ⋅ dx + C0 Function DesolveDesolve Variable OdetypeLaplace transform and inverses in the calculator Laplace Transforms= ∑ c n Fourier seriesFunction Fourier Fourier series for a quadratic functionPage Reference MTH/PROBABILITY.. sub-menu part Chapter Probability DistributionsFactorials, combinations, and permutations − 2... − + 1 = − rRandom numbers MTH/PROB menu part Normal distributionStudent-t distribution F distribution Chi-square distributionChapter Statistical Applications Entering dataCalculating single-variable statistics Page Obtaining frequency distributions Fitting data to a function y = fx 195238095238 + 2.00857242857*X Correlation Covariance Obtaining additional summary statistics Confidence intervals Page Hypothesis testing Page Reference Chapter Numbers in Different Bases Base menuWriting non-decimal numbers HEX DEC OCT BIN Using SD cards ChapterStoring objects in the SD card Purging an object from the SD card Recalling an object from the SD cardLimited Warranty Service HP Invent USA Regulatory information