What is the chance of this dice appearing twice in a row?

Question

3 friends roll a 6-sided dice. What is the probability that at least two of them roll the same number?

Answer 1

Let $X_{i}$ be the number that friend $i$ rolls for $i=1,2,3$. Furthermore, let $A_{1}=\{X_{1}=X_{2}\}$, $A_{2}=\{X_{2}=X_{3}\},$ and $A_{3}=\{X_{1}=X_{3}\}$. Then we seek the probability $P(A_{1} \cup A_{2} \cup A_{3})$. This is computed using the inclusion-exclusion principle as follows $$P(A_{1} \cup A_{2} \cup A_{3})=\\ \sum_{i=1}^{3}P(A_{i})-P(A_{1} \cap A_{2})-P(A_{2} \cap A_{3})-P(A_{1} \cap A_{3})+P(A_1 \cap A_{2} \cap A_{3}).$$ Now, observe that the rolls of the three friends are independent, and that each of $P(A_{i} \cap A_{j})=P(A_{i}\,|\,A_{j})P(A_{j})$ for $1\leq i, j \leq3$ such that $i \neq j$, and $P(A_{1} \cap A_{2} \cap A_{3})=P(A_{1}\,|\,A_{2} \cap A_{3})P(A_{2} \cap A_{3})=P(A_{1}\,|\,A_{2} \cap A_{3})P(A_{3}\,|\,A_{2})P(A_{2}).$ This should give you enough ammunition to finish the problem. Note that, for example, $$P(A_{1})=P(X_1=X_2=x)=P(X_1=x,X_2=x)=P(X_1=x)P(X_2=x)=\left(\frac{1}{6}\right)^2=\frac{1}{36}$$ for $x \in \{1,...,6\}$, where the third equality is by the independence of the rolls of the dice. Key hint: as an instance of conditional events in this context, suppose that the second friend rolls the same number as the third friend. Will this assumption affect the chances that the first friend rolls the same number as the third friend?