I used
this binomial calculator and got a p-value of 0.0726 or 7.26%. I used 6 or higher instead of exactly 6, since you would have found any higher value even more interesting, and also took the liberty of changing p to the more precise value of 0.0933 from 0.0932
.
But the important question is whether you decided to test exactly these parameters before you looked at the data or looked at the data first and then decided to test this particular scenario. If you looked at the data first then you have a problem. There might be tens of different scenarios that you could have found interesting. The probability of this particular scenario happening by random chance may only be 7% but the probability of at least one of the interesting scenarios happening may well be (and in this case I think is) over 50%.
But even if this is what happened all is not lost. You now have a hypothesis that you can test. Find a different sample and repeat the test for the exact same parameters. And here's a beauty: the p-value is not determined by the phenomenon you are looking at. It's dependent on sample size. If you can get N=96 with 18 "successes" (expected if the first sample was typical) you'd get a p-value of 0.003. Just remember to construct your sample blindly, no looking at the data before you got it finalized
In the interest of openness I should state that I don't think there's anything to this. Actually, I suspect Tarnas looked at a large number of different scenarios until he actually found this one, with different planets, different professions etc. If he did this and doesn't admit it in his book he's just a crook. If he didn't he's just very lucky
But in the end if someone believes there might be something to this he can go ahead and test it. As long he's got specific predictions and a large sample that wasn't used to come up with the model he'll get a definite answer one way or the other.