Q. 2.16 - A First Course in Probability

Question

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

$$\begin{gathered} \left(a\right)P(no two alike)=.0926; \\ \left(b\right)P(one pair)=.4630; \\ \left(c\right)P(two pair)=.2315; \\ \left(d\right)P(three alike)=.1543; \\ \left(e\right)P(full house)=.0386; \\ \left(f\right)P(four alike)=.0193; \\ \left(g\right)P(five alike)=.0008. \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) P(no two alike)=0.0926 \\ \left(b\right)P(one pair)=0.4630 \\ \left(c\right) P( two pairs )=0.2315 \\ \left(d\right)P(three alike)=0.1543 \\ \left(e\right) P( Fullhouse )=0.0386 \\ \left(f\right) P( Four alike)=0.0193 \\ \left(g\right)P(Five alike)=0.0008 \end{gathered}$$

Step 1 Given Information.

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

Step 2 Part (a) Explanation.

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

Step 3 Part (b) Explanation.

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

Step 4 Part (c) Explanation.

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

Step 5 Part (d) Explanation.

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

Step 6 Part (e) Explanation.

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

Step 7 Part (f) Explanation.

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

Step 8 Part (g) Explanation.

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