This is one of the following six articles on Solving Problems With Other Distributions in Excel

Solving Uniform Distribution Problems in Excel 2010 and Excel 2013

Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013

Solving Exponential Distribution Problems in Excel 2010 and Excel 2013

Solving Beta Distribution Problems in Excel 2010 and Excel 2013

Solving Gamma Distribution Problems in Excel 2010 and Excel 2013

Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

# Overview of the

Multinomial Distribution

The multinomial distribution is a discrete distribution, not a continuous distribution. This means that the objects that form the distribution are whole, individual objects. This distribution curve is not smooth but moves abruptly from one level to the next in increments of whole units.

Excel does not provide the multinomial distribution as one of its built-in functions. Users must generate their own Excel formulas.

The multinomial distribution provides the probability of a combination of specified outputs for a given number of trials that are totally independent. The probability of each of the individual outputs of each of the trials must be known in order to utilize the multinomial distribution to calculate the probability of that unique combination of outputs occurring in the given trials.

The multinomial distribution utilizes ** Sampling With Replacement**. Each sampled object is placed back into the population before the next sample is taken from the population.

Here is the formula for calculating the probability of a multinomial distribution:

P ( X_{1} = n_{1}, X_{2} = n_{2}, …, X_{k} = n_{k} ) =

= [ (n!) / (n_{1}! * n_{2}! * …*n_{k}!) ] * [Pr(Output_{1})]^{n1} * [Pr(Output_{2})]^{n2} * … * [Pr(Output_{k})]^{nk}

An example makes the multinomial distribution easier to understand. An example follows:

## Multinomial PDF Problem Solved in

Excel

A box contains 5 red marbles, 4 white marbles, and 3 blue marbles. A marble is sampled at random, its color noted, and then the marble IS REPLACED BEFORE THE NEXT SAMPLE IS TAKEN. 6 marbles are sampled in this manner.

Calculate the probability that out of those 6 sampled marbles, 3 are red, 2 are white, and 1 is blue.

Output_{1} = Sampled marble is red

Output_{2} = Sampled marble is white

Output_{3} = Sampled marble is blue

The population consists of 12 total marbles of which 5 are red, 4 are white, and 3 are blue.

Pr(Output_{1}) = 5/12

Pr(Output_{2}) = 4/12

Pr(Output_{3}) = 3/12

n = total number of trials (marbles sampled) = 6

X_{1} = Actual Count of Output_{1} in n trials

X_{2} = Actual Count of Output_{2} in n trials

n_{1} = Hypothetical Count of Output_{1} in n trials = 3

n_{2} = Hypothetical Count of Output_{2} in n trials = 2

n_{3} = Hypothetical Count of Output_{2} in n trials = 1

The probability of 3 red, 2 white, 1 blue = P(3 red, 2 white, 1 blue)

= P ( X_{1} = n_{1}, X_{2} = n_{2}, …, X_{k} = n_{k} )

= P ( X_{1} = 3, X_{2} = 2, X_{3} = 3 )

P ( X_{1} = n_{1}, X_{2} = n_{2}, …, X_{k} = n_{k} ) =

= [ (n!) / (n_{1}! * n_{2}! * …*n_{k}!) ] * [Pr(Output_{1})]^{n1} * [Pr(Output_{2})]^{n2} * … * [Pr(Output_{k})]^{nk}

= [ (6!) / (3! * 2! * 1! ) ] * [5/12]^{3}** *** [4/12]^{2}** *** [3/12]^{1}

= 625 / 5184 = 0.12056 = 12.06%

There is a 12.06 percent probability that out of those 6 sampled marbles, EXACTLY 3 are red, EXACTLY 2 are white, and EXACTLY 1 is blue.

Because the probability of EXACT number of each possible output have been calculated, the multinomial distribution’s PDF (Probability Density Function) has been calculated in this example.

Excel does not provide the multinomial distribution as one of its built-in

functions. Users must generate their own Excel formulas. In this case,

it would be the following:

P( 3 red, 2 white, 1 blue) =

= [ (6!) / (3! * 2! * 1! ) ] * [5/12]^{3}** *** [4/12]^{2 }* [3/12]^{1}

= **( **FACT(6) / **( **FACT(3) * FACT(2) * FACT(1) **) ) ***** (**(5/12)^3**) *** **(**(4/12)^2**) *** **(**(3/12)^1)

= 0.1206 = 12.06%

### Binomial Distribution Is the Multinomial Distribution With k = 2

The multinomial distribution is a generalization of the binomial distribution. When k = 2, the multinomial distribution is the binomial distribution. The following example demonstrates this:

Calculate the probability that 15 flips of a fair coin (p = 0.5) will produce EXACTLY 4 heads (and therefore EXACTLY 11 tails).

The flipping of a coin is considered sampling with replacement because the probability of obtaining a specific output (heads for example) remains constant during every trial.

Solve with the binomial distribution’s PDF in Excel as follows:

X = 4

n = 15

p = 0.5

Pr(X=4;n=15,p=0.5)

= BINOM.DIST(X,n,p,FALSE) = BINOM.DIST(4,15,0.5,FALSE) = 0.0417

Solve with the multinomial distribution’s PDF in Excel as follows:

Output_{1} = Heads from a single trial (coin flip)

Output_{2} = Tails from a single trial

Pr(Output_{1}) =0.5

Pr(Output_{2}) =0.5

n = total number of trials (coin flips) = 15

X_{1} = Actual Count of Output_{1} in n trials

X_{2} = Actual Count of Output_{2} in n trials

n_{1} = Hypothetical Count of Output_{1} in n trials = 4

n_{2} = Hypothetical Count of Output_{2} in n trials = 11

P ( X_{1} = n_{1}, X_{2} = n_{2}, …, X_{k} = n_{k} ) =

= [ (n!) / (n_{1}! * n_{2}! * …*n_{k}!) ] * [Pr(Output_{1})]^{n1} * [Pr(Output_{2})]^{n2} * … * [Pr(Output_{k})]^{nk}

= [ (15!) / (4! * 11!) ] * [0.5]^{4}** *** [0.5]^{11}

= 0.0417

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
- One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
- Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- 1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- 2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression in 6 Steps in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Randomized Block Design ANOVA in Excel
- Repeated-Measures ANOVA in Excel
- Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013
- Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013
- Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
- Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013

- ANCOVA in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel

- Post Hoc Testing in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers

- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables and Pivot Charts
- SEO Functions in Excel
- Time Series Analysis in Excel
- VLOOKUP
- Simplifying Useful Excel Functions

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