when would next integer $n+1$ be a prime in a given range of $p_*< p< p_*^2$?

Question

Conjecture

that along the sequence of natural numbers $n\in\Bbb N$, if walking upwards $1,2,3,4,\ldots,n,n+1,\ldots,$ from every integer to the next (starting with $n=1$), the probability $\phi_p$ that every next integer $n+1$ would be a prime in a given range of $p_*< p< p_*^2$, equals:

$$\Phi_{p_*}=\prod_{p=2}^{p_*} \phi^{(\text{Markov }1)}_p=\prod_{p=2}^{p_*} \frac{p-2}{p-1}$$

(transition probability in accordance to a Markov chain first order (indexed Markov 1); and taking into account our knowledge of the current state $n$ before we walk to $n+1$).

Question: is the above conjecture mathematically correct?

PS: the above is related to the post and post.

Answer 1

You could not have a fixed transition probability "walking upwards" in a given range $p_*<p<p_*^2$ since these intervals are overlapping. For example, for $n$ in $31,32,32,\ldots,100$ you would have the probability equal to $\Phi_{11},\Phi_{13},\Phi_{17}$ and others even though those differ.

It's not clear what you mean by "probability" here. This doesn't look like a Markov process. When $n=6k+1$, "taking into account our knowledge of the current state $n$" we can say that the probability that $n+1,n+2,n+3$ are prime is zero.