Probability In Unexpected Places

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.

djlemahieu
DJ LeMahieu, one of only two MLB players to hit close to 0.350 in 2016 (he hit 0.348), had 69 four-at-bat games in the 146 games he played. He went ¾, 8.7 % of the time . His teammate Gerardo Parra who hit 0.253 did it only in 2 of 52 four-at-bat-games, or 3.8% of the time. The ratio was 2.26 for the two players , very close to the theoretical expectation of 2.27.

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?

hundsrule

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.)

8c9690fdfce673e0d5c702085bf53400A 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.

 

Science Dimensions On A Baseball Diamond

Atomic and astronomical scales are not in the least intuitive. Powers of 10 tables have often been used to contrast the size of stars with the radii of atoms, but since spring training has arrived, why not use the baseball diamond to mentally animate some of the baffling dimensions of science?

dodger
 * Obviously the 1 mm proton analogy is not related to the 1 cm Earth!

If you reduce the Earth to the size of an eraser (1 cm) at the end of a pencil at home plate, the sun (109.7 times wider) becomes a 1.1 meter beach ball, 117 meters away. That would be about a meter past the left or right center field wall at Dodger Stadium. Interestingly, in this model,  the next nearest star would still be triple the distance (31 000 km) between Alaska and India. From a mass perspective, if the earth’s mass is reduced to that of a baseball, the sun would weigh as much as 644 spectators, averaging 75 kg each.

If you turn the pencil around and make a one millimeter dot on home plate to represent a proton*, well, if you scale up such a small particle to those dimensions, you will find a hydrogen  atom’s lone electron past first base, 35.3 meters away. This reveals what Rutherford demonstrated over a century ago, that an atom is mostly empty space, just like the baseball field is mostly empty if fielders are not mobile. If the mass of the electron is scaled up to that of a baseball, then the proton, which is 1836 times more massive, acquires the mass of all the bats in the clubhouse (285).

What if the Milky Way galaxy is scaled down to the size of Dodger Stadium? How small does the Earth become? Well, the Earth would only be 6 X 10-12 m wide, about 10 times smaller than the smallest atom. How small is a baseball player rendered? To about the upper limit for the size of a quark. But we’re getting too abstract. If we scale Prince Fielder’s mass down to that of an H1V virus, how massive does the Earth become? As massive as a blue whale.

 

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