A LOT of people try to calculate something that should intrinsically defy probability--a singular instance--like the universe's beginning.
Penrose is not a theist either, instead having his own theory of cyclical time, which we'll get to later.
Penrose's observation about a small low-entropy region being more probable than the highly-ordered observable universe means that if there IS a hypothetical Creator, His/Her intent is not to create US--not just one particular world, one intelligent species--but to allow many such worlds to come about.
Proton mass to Planck mass: for a star to live more than a billion years means the ratio must be 10 to the minus nineteenth. So the odds of that are 10 to the nineteenth power.
The electron and the neutron must match each other's mass to an accuracy of 10 to the negative 22 in planck units, so the probably of that is 10 to the 22nd power. Same is true of the neutrinoes. The chances of that in three rolls of the dice are 10 to the 66nd power.
Together, that makes 10 to the 85th power.
Cosmological constant---for the universe to live long enough to form galaxies, the cosmological constant must be 10 to the minus sixtieth power in Planck units. The probability of getting that randomly is 10 to the sixtieth power. Multiplying times the previous one gives us one chance in 10 to the 145th power.
Strengths of the four main forces: range from strong nuclear force to weak and electromagnetic force, is one to 100 in ratio. That's 1 part in 100.
Range of those forces. Some range the entire universe, like electromagnetism. The weak or nuclear forces only go as far as the nucleus. The ratio of these forces or at least 1 in 10 to the 40th power.
The chances of a world like ours being picked by just random assignment of properties to these forces and constants to create a universe lasting billions of years and nuclear and atomic physics more or less like ours--is one to 10 to the 229th power.
Here's a link to Penrose's entropy estimate and here's a link to Paul Davies' virtual reality multiverse matrix idea.
NEXT: High Probability.