But are Fetch and Fitch also Dalmations?

I have three dogs: Fatch, Fetch and Fitch. Yesterday, I found a brown 12-inch stick for them to play with. I marked the top and bottom of the stick and then threw it for Fatch. Fatch, a Dalmatian, bit it in a random spot — leaving a mark — and returned it to me. In her honor, I painted the stick black from the top to the bite and white from the bottom to the bite.

I subsequently threw the stick for Fetch and then for Fitch, each of whom retrieved the stick by biting a random spot. What is the probability that Fetch and Fitch both bit the same color (i.e., both black or both white)?

Let's denote the random spot (measured in feet from the bottom of the stick) where Fatch the Dalmatian and Fetch and Fitch of unknown breed bit the stick as $A$, $E$ and $I$, where each can now be modeled as i.i.d. uniform random variables on $[0,1].$

Once Fatch has returned the stick with say $A = a$ for some $a \in [0,1],$ by the independence of $E$ and $I,$ we can calculate \begin{align*}\mathbb{P} \{ \text{same color} \mid A = a \} &= \mathbb{P} \{ E, I \in [0,a] \} + \mathbb{P} \{E, I \in [a,1]\} \\ &= \mathbb{P} \{ E \in [0,a] \} \mathbb{P} \{ I \in [0,a] \} + \mathbb{P} \{ E \in [a,1] \} \mathbb{P} \{ I \in [a,1] \}\\ &= a^2 + (1-a)^2.\end{align*} So if we integrate over all the possible Fatch the Dalmatian bite marks, we get the probability that Fetch and Fitch bite the same colored portion of the stick as $$\mathbb{P} \{ \text{same color} \} = \int_0^1 \mathbb{P} \{ \text{same color} \mid A = a \} \, da = \int_0^1 a^2 + (1-a)^2 \,da = \frac{2}{3}.$$ However, we are still no closer to divining the breed of Fetch and Fitch.

If I were to get more dogs of unknown parentage, say $n,$ then (a) I would likely eventually run of out vowels in order to fit the naming F*tch naming convention, but also (b) end up with a probability that all of then would bite in the same color of $$\mathbb{P} \{ n \,\text{bites in same color} \} = \int_0^1 a^n + (1-a)^n \,da = \frac{2}{n+1}.$$ If I went full Cruella and secured $100$ dogs of unknown breed and $1$ Dalmatian, the probability that all $100$ dogs after Fatch would bite in the same color would be a not insignificant $\frac{2}{101} = 1.98\%.$