0 Probability

The the­o­ret­ical for­malism of con­tem­po­rary physics is a prob­a­bility cal­culus. The prob­a­bility algo­rithms it places at our dis­posal — state vec­tors, wave func­tions, den­sity matrices, sta­tis­tical oper­a­tors, and what have you — all serve the same pur­pose, which is to cal­cu­late the prob­a­bil­i­ties of pos­sible mea­sure­ment out­comes on the basis of actual mea­sure­ment out­comes. That’s reason enough to put together what we already know and what we need to know about probabilities.

Prob­a­bility is a mea­sure of like­li­hood ranging from 0 to 1. If an event has a prob­a­bility equal to 1, it is cer­tain that it will happen; if it has a prob­a­bility equal to 0, it is cer­tain that it will not happen; and if it has a prob­a­bility equal to 12, then it is as likely as not that it will happen.

Tossing a fair coin yields heads with prob­a­bility 12. Casting a fair die yields any given nat­ural number between 1 and 6 with prob­a­bility 16. These are exam­ples of the prin­ciple of indif­fer­ence, which states: if there are n mutu­ally exclu­sive and jointly exhaus­tive pos­si­bil­i­ties (pos­sible events), and if we have no reason to con­sider any one of them more likely than any other, then each pos­si­bility should be assigned a prob­a­bility equal to 1/​n.

(Saying that events are mutu­ally exclu­sive is the same as saying that at most one of them hap­pens. Saying that events are jointly exhaus­tive is the same as saying that at least one of them hap­pens. Since one of them is sure to happen, the n prob­a­bil­i­ties must add up to 1. Since the prin­ciple requires that the indi­vidual prob­a­bil­i­ties be equal, each prob­a­bility must be 1/​n.)

blank

Sub­jec­tive vs. objec­tive probabilities

There are two kinds of sit­u­a­tions in which we may have no reason to con­sider one pos­si­bility more likely than another. In sit­u­a­tions of the first kind, there are objec­tive mat­ters of fact that would make it cer­tain, if we knew them, that a par­tic­ular event will happen, but we don’t know any of the rel­e­vant mat­ters of fact. The prob­a­bil­i­ties we assign in this case, or when­ever we know some but not all rel­e­vant facts, are in an obvious sense sub­jec­tive. They are igno­rance prob­a­bil­i­ties. They have every­thing to do with our (lack of) knowl­edge of rel­e­vant facts, but nothing with the exis­tence of rel­e­vant facts. There­fore they are also known as epis­temic prob­a­bil­i­ties.

In sit­u­a­tions of the second kind, there are no objec­tive mat­ters of fact that would make it cer­tain that a par­tic­ular event will happen. There may not even be any objec­tive matter of fact that would make it more likely that one event will occur rather than another. There isn’t any rel­e­vant fact that we are igno­rant of. The prob­a­bil­i­ties we assign in this case are nei­ther sub­jec­tive nor epis­temic. They have every right to be con­sid­ered objec­tive. Quantum-​​mechanical prob­a­bil­i­ties are essen­tially of this kind.

blank

Prob­a­bil­i­ties and rel­a­tive frequencies

Until the advent of quantum mechanics all prob­a­bil­i­ties were thought to be sub­jec­tive. This had two unfor­tu­nate con­se­quences. The first is that prob­a­bil­i­ties came to be thought of as some­thing intrin­si­cally sub­jec­tive. The second is that some­thing that was not a prob­a­bility at all but a rel­a­tive fre­quency — came to be referred to as an “objec­tive probability.”

Rel­a­tive fre­quen­cies are useful in that they allow us to mea­sure the like­li­hood of pos­sible events, at least approx­i­mately, pro­vided that trials can be repeated under con­di­tions that are iden­tical in all rel­e­vant respects. We obvi­ously cannot mea­sure the like­li­hood of heads by tossing a single coin. But since we can toss a coin any number of times, we can count the number NH of heads and the number NT of tails obtained in N tosses and cal­cu­late the frac­tion NH/​N of heads and the frac­tion NT/​N of tails. And we can expect the mag­ni­tude of the dif­fer­ence NH − NT to increase sig­nif­i­cantly slower than the sum N = NH + NT, so that the mag­ni­tude of the dif­fer­ence NH/​N − NT/​N approaches 0 as N “goes to infinity”:

|NH/​N − NT/​N| → 0   as N → ∞.

We can there­fore expect the rel­a­tive fre­quen­cies NH/​N and NT/​N to approach the respec­tive prob­a­bil­i­ties pH and pT in this “limit”:

NH/​N → pH and   NT/​N → pT as N → ∞.

blank

Adding and mul­ti­plying probabilities

Sup­pose you roll a die, and sup­pose you win if you throw either a 1 or a 6 (no matter which). Since there are six equiprob­able out­comes, two of which make you win, your chances of win­ning are 26. In this example it is appro­priate to add probabilities:

p(1 v 6) = p(1) + p(6).

The gen­eral rule is:

Sum Rule. Given n mutu­ally exclu­sive and jointly exhaus­tive events (such as the pos­sible out­comes of a mea­sure­ment), and given m of these n events, the prob­a­bility that one of the m events takes place (no matter which) is the sum of the indi­vidual prob­a­bil­i­ties of these m events. (One nice thing about rel­a­tive fre­quen­cies is that they make a rule such as this vir­tu­ally self-​​evident.)

Sup­pose now that you roll two dice. And sup­pose that you win if your total equals 12. Since there are now 6×6 equiprob­able out­comes, only one of which makes you win, your chances of win­ning are 136. In this example it is appro­priate to mul­tiply probabilities:

p(6 & 6) = p(6) × p(6).

The gen­eral rule is:

Product rule. The joint prob­a­bility p(e1&…&em) of m inde­pen­dent events e1,…,em (that is, the prob­a­bility with which all of them happen) is the product p(e1)×…×p(em) of the prob­a­bil­i­ties of the indi­vidual events.

It must be stressed that the product rule only applies to inde­pen­dent events. Saying that two events e1 and e2 are inde­pen­dent is the same as saying that the prob­a­bility of e1 is inde­pen­dent of whether or not e2 hap­pens, and vice versa.

blank

Con­di­tional prob­a­bil­i­ties and correlations

If e1 and e2 are not inde­pen­dent, one has to dis­tin­guish between mar­ginal prob­a­bil­i­ties, which are assigned to either event regard­less of whether the other event hap­pens, and con­di­tional prob­a­bil­i­ties, which are assigned to either event depen­dent on the out­come of the other event. If the two events are not inde­pen­dent, their joint prob­a­bility is given by

p(e1 & e2) = p(e1|e2) p(e2) = p(e2|e1) p(e1),

where p(e1) and p(e2) are mar­ginal prob­a­bil­i­ties, while p(e1|e2) is the prob­a­bility of e1 con­di­tional on the occur­rence of e2, and p(e2|e1) is the prob­a­bility of e2 con­di­tional on the occur­rence of e1.

e1 and e2 are said to be cor­re­lated if (and only if)

p(e1|e2) ≠ p(e1|e2),

where p(e1|e2) is the prob­a­bility with which e1 takes place if e2 does not take place. Saying that the two events are inde­pen­dent is thus the same as saying that they are uncor­re­lated, for

p(e1 & e2) = p(e1) × p(e2)

holds if and only if

p(e1|e2) = p(e1|e2)

holds, in which case both sides equal the mar­ginal prob­a­bility p(e1).

Next