When I was young my Mom would say that you often find things when you’re not looking for them. It took me longer than it should have to realise that the truism is based on probability. We typically look for a lost item only briefly after we’ve lost it. Or occasionally we decide to look for objects that someone else may have lost in the sand. But for those of us who are not professional treasure hunters, we spend most of our days preoccupied with other goals. Yet for the bulk of the time when we are not actively seeking, our eyes are still open to the possibility of finding something. So it’s more probable that we stumble upon a lost item when we have time on our side.
Few of us think of everyday things mathematically and out of the few who do, fewer still apply the often non-intuitive formulas of probability. How many of us would guess that in baseball, a 0.350 hitter (someone who gets a hit in 35% of his at bats) is more than twice as likely to get 3 hits out of 4 at bats when compared to a player who only hits 0.250? To calculate each probability we have to first get the possible combinations of outs and hits in 4 at bats which amounts to dividing the permutations (4 X 3 X 2) by the number of permutations per combination(3 X 2 X 1). This yields 4. In turn that number has to be multiplied by the probability of getting a hit raised to the power of hits and also multiplied by the probability of not getting a hit, raised by the difference between at bats and hits. A mouthful indeed!
For the 0.350 hitter, P(x) = (4 X 3 X 2)/(3 X 2 X 1)*0.3503 * 0.6501 = 0.111, but for the 0.250 hitter, P(x) = (4 X 3 X 2)/(3 X 2 X 1)*0.2503 * 0.7501 = 0.0469. Since 0.111/0.0469 =2.38, the 0.350 hitter is more than twice as likely to go 3 for 4 in a game.
As we move from the Newtonian level of large bats making contact with baseballs to the sublime level of quantum mechanics, we still encounter probability. Why for instance is atomic nitrogen’s electron configuration like the one indicated with the green tick mark?
Why aren’t the first two electrons at the 2p level paired up in the same orbital? In the first setup, repulsion among electrons in a single orbital is avoided. But in the actual configuration something else is going on as well . As such, each electron can exchange places with any of the two other electrons since all three have the same spin, so we have a total of 6 permutations.
The incorrect setup would imply that the lone electron could only exchange with one other electron from the filled orbital since only one of those would have the same spin. The multiple possibilities of the first scenario— the one following the first of so-called Hund’s rules—stabilise the atom and is the one actually found in nature. (For a more rigorous treatment and to see how Hund’s first rule does not always apply to molecules, consult this review paper.)
A simpler example of probability at work in the realm of chemistry occurs when a drop of vegetable colouring is added to a glass of undisturbed water. The molecules of dye are in constant motion. Those within the drop are more likely to collide with themselves than with water. But eventually, those self-collisions will lead molecules away from the pack and towards the edge of the drop. There, they will be closer to moving water molecules. Collisions with water will divert the dye towards areas where there are more water molecules than dye, increasing the probability that subsequent encounters will be with other molecules of the universal solvent. Since that original drop represented just one small zone out of the total volume of solution, it becomes extremely unlikely that the dye will reconvene to its point of origin. (Unless of course we evaporate the water!)
So without stirring, the dye will spread uniformly and dissolve thanks to the heat in the environment that powers their motion, and thanks of course to probability.