So, I was curious what the probability of someone passing the PE exam based on completely random guessing (because it seemed feasible), so, I did some math. The probabilities are based on an assumed passing score of 70% (56/80), using the binomial cumulative probability... that is, for randomly guessing all 80 questions, what's the probability of guessing at least 56 correct? Thus, calculate the binomial cdf for (n=80, p=0.25, and x=56 + x=57 + ...x=80)
For all 80 questions, the probability of guessing at least 56 correctly turns out to be 3.64 x 10^-17. That is, 1 in 27,500,000,000,000,000 test takers will be likely to pass the exam with random guessing.
However, if a test taker is 100% confident that he worked out half the questions correctly, and wanted to simply guess on the other 40 questions, he would only need to guess 16 problems correctly out of the remaining 40. The probability is calculated similarly, with a binomial cdf of (n=40, p=0.25, and x=16 + x=17....x=40). This turns out to be a probability of 0.026. That is, 1 out of about 50 test takers who definitely answer 40 questions correctly can expect to pass the exam by guessing on the remaining questions.
Here's the breakdown:
0 definitely correct, Guess 56/80 correctly to pass : 1 in 27,500,000,000,000,000 will pass(1 out of 27 Quintillion)
10 correct, Guess 46/70: 1 in 1,170,000,000,000 will pass (1 out of a trillion)
20 correct, Guess 36/60: 1 in 103,000,000 will pass
30 correct, Guess 26/50: 1 in 26,300 will pass
40 correct, Guess 16/40: 1 in 50 will pass.
50 correct, Guess 6/30, 4 in 5 will pass.
60 correct ... pretty sure you passed!