Question

Poker dice are played by simultaneously rolling $5$dice. Show that

$$\begin{gathered} \left(a\right)P(notwoalike)=.0926; \\ \left(b\right)P(onepair)=.4630; \\ \left(c\right)P(twopair)=.2315; \\ \left(d\right)P(threealike)=.1543; \\ \left(e\right)P(fullhouse)=.0386; \\ \left(f\right)P(fouralike)=.0193; \\ \left(g\right)P(fivealike)=.0008. \end{gathered}$$

Short answer

$$\begin{gathered} \left(a\right)P(notwoalike)=0.0926 \\ \left(b\right)P(onepair)=0.4630 \\ \left(c\right)P(twopairs)=0.2315 \\ \left(d\right)P(threealike)=0.1543 \\ \left(e\right)P(Fullhouse)=0.0386 \\ \left(f\right)P(Fouralike)=0.0193 \\ \left(g\right)P(Fivealike)=0.0008 \end{gathered}$$

Step-by-step solution

Given Information.

Poker dice are played by simultaneously rolling $5$dice.

Part (a) Explanation.

$P\left(\text{no two alike}\right)=\frac{6\times 5!}{6^5}=0.0926$.

Part (b) Explanation.

$P\left(\text{one pair}\right)=\frac{\left(\begin{array}{c}5\\ 2\end{array}\right)\left(\begin{array}{c}6\\ 1\end{array}\right)\left(\begin{array}{l}5\\ 3\end{array}\right)3!}{6^5}=0.4630$.

Part (c) Explanation.

$P\left(\text{two pairs}\right)=\frac{\left(\begin{array}{l}5\\ 2\end{array}\right)\left(\begin{array}{l}3\\ 2\end{array}\right)\left(\begin{array}{l}6\\ 2\end{array}\right)\left(\begin{array}{l}4\\ 1\end{array}\right)}{6^5}=0.2315$.

Part (d) Explanation.

$P\left(\text{three alike}\right)=\frac{\left(\begin{array}{c}5\\ 3\end{array}\right)\left(\begin{array}{l}6\\ 1\end{array}\right)\left(\begin{array}{l}5\\ 2\end{array}\right)2!}{6^5}=0.1543$.

Part (e) Explanation.

$P\left(\text{Fullhouse}\right)=\frac{\left(\begin{array}{l}5\\ 3\end{array}\right)\left(\begin{array}{l}6\\ 1\end{array}\right)\left(\begin{array}{l}5\\ 1\end{array}\right)}{6^5}=0.0386$.

Part (f) Explanation.

$P\left(\text{Four alike}\right)=\frac{\left(\begin{array}{l}5\\ 4\end{array}\right)\left(\begin{array}{l}6\\ 1\end{array}\right)\left(\begin{array}{l}5\\ 1\end{array}\right)}{6^5}=0.0193$.

Part (g) Explanation.

$P\left(\text{Five alike}\right)=\frac{\left(\begin{array}{l}6\\ 1\end{array}\right)}{6^5}=0.0008$.