What is the expected value of the largest of the three dice rolls?

More generally let's find $E[X_\max]$ and $E[X_\min]$ where $X_\max$ and $X_\min$ are the largest and smallest outcomes among $n$ independent rolls of an $s$-sided die with sides numbered from $1$ to $s$.

Define $X_i$ to be the random variable whose value is $1$ if $X_\max\ge i$ and $0$ otherwise; then $$X_\max=\sum_{i=1}^sX_i$$ and $$E[X_\max]=E\left[\sum_{i=1}^sX_i\right]=\sum_{i=1}^sE[X_i]=\sum_{i=1}^sP(X_i=1)=\sum_{i=1}^sP(X_\max\ge i)$$ $$=\sum_{i=1}^s[1-P(X_\max\lt i)]=\sum_{i=1}^s\left[1-\left(\frac{i-1}s\right)^n\right]=s-s^{-n}\sum_{i=1}^s(i-1)^n=\boxed{s-s^{-n}\sum_{i=1}^{s-1}i^n}.$$ Since the outcome of a single roll is distributed symmetrically about its mean $\frac{1+s}2$, we have $E[X_\min]+E[X_\max]=1+s$, so that $$E[X_\min]=1+s-E[X_\max]=\boxed{1+s^{-n}\sum_{i=1}^{s-1}i^n}.$$

When $s=6$ and $n=3$ we have $$E[X_\max]=6-6^{-3}(1^3+2^3+3^3+4^3+5^3)=6-6^{-3}\binom62^2=6-\frac{25}{24}=\boxed{\frac{119}{24}}$$ and $$E[X_\min]=7-E[X_\max]=7-\frac{119}{24}=\boxed{\frac{49}{24}}.$$

Let $X_{i}$ be the value from individual die

Let Z be the max value: $Z = max(X_1, X_2, X_3)$

The CDF of Z is $F_Z(x) = P(Z\le x) = P(X_1\le x)P(X_2\le x) P(X_3\le x) = F_{X_1}(x)^3 = (x/6)^3 $

Therefore PMF of Z: $f_Z(x) = F_Z(x) - F_Z(x-1) = (x^3-(x-1)^3)/6^3 $

$ E(Z) = \sum_{x=1}^6 xf_Z(x) = 119/24 $

Let X denote the largest value, then:

• $P(X=1)=\frac{ 1}{216}$
• $P(X=2)=\frac{ 7}{216}$
• $P(X=3)=\frac{19}{216}$
• $P(X=4)=\frac{37}{216}$
• $P(X=5)=\frac{61}{216}$
• $P(X=6)=\frac{91}{216}$

Hence the expected value is:

$$1\cdot\frac{1}{216}+2\cdot\frac{7}{216}+3\cdot\frac{19}{216}+4\cdot\frac{37}{216}+5\cdot\frac{61}{216}+6\cdot\frac{91}{216}=\frac{119}{24}$$

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Details:

Let $C_n$ denote the number of combinations with largest value $n$.

Observe that $C_n=n^3-\sum\limits_{k=1}^{n-1}C_{n-1}$, therefore:

• $C_1=1^3=1$
• $C_2=2^3-1=7$
• $C_3=3^3-1-7=19$
• $C_4=4^3-1-7-19=37$
• $C_5=5^3-1-7-19-37=61$
• $C_6=6^3-1-7-19-37-61=91$