19.2: Distribution of Results for Multiple Trials with Three Possible Outcomes

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Let us extend the ideas we have developed for binomial probabilities to the case where there are three possible outcomes for any given trial. To be specific, suppose we have a coin-sized object in the shape of a truncated right-circular cone, whose circular faces are parallel to each other. The circular faces have different diameters. When we toss such an object, allowing it to land on a smooth hard surface, it can wind up resting on the big circular face (\(\boldsymbol{H}\)eads), the small circular face (\(\boldsymbol{T}\)ails), or on the conical surface (\(\boldsymbol{C}\)one-side). Let the probabilities of these outcomes in a single toss be \(P_H\), \(P_T\), and \(P_C\), respectively. In general, we expect these probabilities to be different from one another; although, of course, we require \(1=\left(P_H+P_T+P_C\right)\).

1024px-CroppedCone.svg.png — Figure 1: A truncated right-circular cone (CC BY-SA Unported; Dirk Hünniger via Wikipedia).

Following our development for the binomial case, we want to write an equation for the total probability sum after \(n\) tosses. Let \(n_H\), \(n_T\), and \(n_C\) be the number of \(H\), \(T\), and \(C\) outcomes exhibited in \(n_H+n_T+n_C=n\) trials. We let the probability coefficients be \(C\left(n_H,n_T,n_C\right)\). The probability of \(n_H\), \(n_T\), \(n_C\) outcomes in \(n\) trials is

\[C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]

and the total probability is

\[1={\left(P_H+P_T+P_C\right)}^n=\sum_{n_H,n_T,n_C}{C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C} \nonumber \]

where the summation is to be carried out over all combinations of integer values for \(n_H\), \(n_T\), and \(n_C\), consistent with \(n_H+n_T+n_C=n\).

To find \(C\left(n_H,n_T,n_C\right)\), we proceed as before. We suppose that one of the terms with \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides is

\[\left(P_{H,a}P_{H,b}\dots P_{H,f}\right)\left(P_{T,g}P_{T,h}\dots P_{T,m}\right)\left(P_{C,p}P_{C,q}\dots P_{C,z}\right) \nonumber \]

where there are \(n_H\) indices in the set \(\{a,\ b,\ \dots ,\ f\}\), \(n_T\) indices in the set \(\{g,\ h,\ \dots ,\ m\}\), and \(n_C\) indices in the set \(\mathrm{\{}\)p, q,…, z\(\mathrm{\}}\). There are \(n_H!\) ways to order the heads outcomes, \(n_T!\) ways to order the tails outcomes, and \(n_C!\) ways to order the cone-sides outcomes. So, there are \(n_H!n_T!n_C!\) possible ways to order \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides. There will also be \(n_H!n_T!n_C!\) indistinguishable permutations of any combination (particular assignment) of \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides. There are \(n!\) possible permutations of \(n\) probability factors and \(C\left(n_H,n_T,n_C\right)\) distinguishable combinations with \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides. As before, we have

total number of permutations = (number of distinguishable combinations)\({}_{\ }\)\({}_{\times }\) (number of indistinguishable permutations for each distinguishable combination)

so that

\[n!=n_H!n_T!n_C!C\left(n_H,n_T,n_C\right) \nonumber \]

and hence, \[C\left(n_H,n_T,n_C\right)=\frac{n!}{n_H!n_T!n_C!} \nonumber \]

Equivalently, we can construct a sum of terms, \(S\), in which the terms are all of the \(n!\) permutations of \(P_{H,r}\) factors for \(n_H\) heads, \(P_{T,s}\) factors for \(n_T\) tails, and \(P_{C,t}\) factors for \(n_C\) cone-sides. The value of each term in \(S\) will be \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\). Thus, we have

\[S=n!P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]

\(S\) will contain all \(C\left(n_H,n_T,n_C\right)\) of the distinguishable combinations \(n_H\) heads, \(n_T\) tails, and \(n_C\) cone-sides outcomes that give rise to \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\)-valued terms. Moreover, \(S\) will also include all of the \(n_H!n_T!n_C!\) indistinguishable permutations of each of these \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\)-valued terms, and we also have

\[S=n_H!n_T!n_C!C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]

Equating these two expressions for S gives us the number of \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\)-valued terms in the total-probability product,\(\ C\left(n_H,n_T,n_C\right)\). That is,

\[S=n!P^{n_H}_HP^{n_T}_TP^{n_C}_C=n_H!n_T!n_C!C\left(n_H,n_T,n_C\right)P^{n_H}_HP^{n_T}_TP^{n_C}_C \nonumber \]

and, again, \[C\left(n_H,n_T,n_C\right)=\frac{n!}{n_H!n_T!n_C!} \nonumber \]

In the special case that \(P_H=P_T=P_C={1}/{3}\), all of the products \(P^{n_H}_HP^{n_T}_TP^{n_C}_C\) will have the value \({\left({1}/{3}\right)}^n\). Then the probability of any set of outcomes, \(\{n_H,n_T,n_C,\}\), is proportional to \(C\left(n_H,n_T,n_C\right)\) with the proportionality constant \({\left({1}/{3}\right)}^n\).