Microsoft was never a penny stock. It IPO'ed March 13th 1986 for 21$ a share, but like a lot of IPO's opened higher at $25.50.
That fallacy comes from not being able to read a split adjusted chart. Had MSFT never split, current price would be 8K a share. It's market cap on IPO (by Goldman Sachs, btw) was over 770 million, it had over 170 million in revenue annually, 57 million in profit and 11 years of profitability hardly making it comparable to pinky microcaps.
The solution to the game is (23/40).
E(R) = expected value of run = .5p + .8(1 - p) = -.3p + .8
E(P) = expected value of pass = .5p + .2
Set E(P) = E(R)
and you have
-.3p + .8 = .5p + .2
.8p = .6
p = 3/4
E(P) = E(R) = 23/40 if you use either equation.
Note p is a probability you are solving for (what probability the defense should call a run) while capital P stands for PASS. The two p's have nothing to do with each other. Just an unfortunate aspect of probability theory and a problem with the word PASS in it.
What does this mean? The offense, which calls the play, should randomly select run 3 times out of 4, and pass 1 times out of 4.
Solving for q, what probability the defense should play run looks like this.
E(R) = .5q +.7(1 - q)
E(p) = .8q + .2(1 -q)
E(R) = E(P)
.5q +.7(1 - q) = .8q +.2(1 - q)
-.2q +.7 = .6q +.2
.5 = .8q
q = 5/8
So the defense should play run 5/8 of the time. E(P) = E(R) = 23/40 again, of course.
This is assuming both coaches understand game theory. :) Not surprising that on third and short, with both teams deploying optimum strategies, the offense converts 57.5% and defense stops 42.5%.
So, do I have your undying love? Oh, the answer is 23/40, in case it got lost in all the proof of work.