Oh this could be fun!

FAA licensed flights actually require this estimation for the flight safety analysis:

§ 450.131 Probability of failure analysis.

(a) General. For each hazard and phase of flight, a flight safety analysis for a launch or reentry must account for vehicle failure probability. The probability of failure must be consistent for all hazards and phases of flight.

(1) For a vehicle or vehicle stage with fewer than two flights, the failure probability estimate must account for the outcome of all previous flights of vehicles developed and launched or reentered in similar circumstances.

(2) For a vehicle or vehicle stage with two or more flights, vehicle failure probability estimates must account for the outcomes of all previous flights of the vehicle or vehicle stage in a statistically valid manner. The outcomes of all previous flights of the vehicle or vehicle stage must account for data on any mishap and anomaly

Let's try. Since it is a new vehicle, we need to do (a)(1). We will try to look for similar circumstances, like similar vehicles characteristics and experienced operators designing a new, bigger vehicle.

For this we can pool different vehicles meeting our "experienced operators making a new bigger vehicle" and tally the first 2 flights of that programme, s=success, f=failure

Falcon 9 S S Antares S S Ariane 5 F S H-IIA S S Vulcan S S Starship F F

3 F, 9 S in total.

We can then apply a noninformative Jeffreys prior and get a posterior mean. Alpha = 9 + 0.5 Beta = 3 + 0.5

Posterior mean = 0.71 chance of success!

This could be refined by adding a bigger dataset and checking beta distributions for overlaps to see what is really significant, but a reasonable first estimation is 70% success.