Problem 2

Billionaires Barry and Bruce decide to justify their worth by simultaneously launching themselves into space. Both hire equally skilled teams of rocket scientists, who can design rockets that reach an altitude $X$ for any $X>0$, but with probability $p(X)$ that for each launch the rocket fails on the launchpad and goes nowhere at all. This failure probability is an increasing function of $X$, with $p(0)=0$ and $p(X)$ approaching 1 as $X$ approaches 100km.

Whoever launches themselves to a greater altitude declares themselves the winner. If both rockets fail, both will try again the next day, until at least one launch succeeds. Both billionaires decide in secrecy on the altitude that they will each design their rocket for.

However, Bruce has become aware that a spy has infiltrated his organisation, and this spy is relaying information about his rocket to Barry! Barry can therefore find out Bruce's chosen altitude ahead of time and design his rocket accordingly, and Bruce is well aware of this.

If both billionaires act optimally (which, in their minds, means acting to maximise their chances of winning), what is Bruce's chance of reaching the higher altitude?

Give your answer rounded to three decimal places.