12:11:14 From Casey Davis : Greetings and welcome back to 7A workshop!
12:11:18 From Casey Davis : How has class been recently?
12:11:47 From Kimia Tavassoli : we just started discussing macro/microstates in DL and i’m pretty confused on how the probability and everything works
12:12:41 From Jessica Marelli : yes same I am confused about the FNT 15 question 3
12:12:45 From Casey Davis : Okay; we can do some probability review.
12:12:56 From Casey Davis : What's the question#3 here?
12:14:01 From Casey Davis : Could you upload the pdf or send it to my email?
12:14:08 From Casey Davis : (casdav@ucdavis.edu)
12:14:16 From Jessica Marelli : yes give me a sec
12:14:20 From Casey Davis : Thanks.
12:14:44 From Casey Davis : Also, are there any questions you'd like to discuss about the PV-graphs or related processes before we move on to probability and entropy?
12:15:33 From Jessica Marelli : Ok I just sent it via email
12:15:53 From Casey Davis : Got it; thanks.
12:16:41 From Casey Davis : So what do you know about microstates and macrostates so far?
12:17:01 From Jessica Marelli : we just learned an example with flipping a coin
12:17:13 From Jessica Marelli : but I was’t really sure how it relates
12:17:17 From Kimia Tavassoli : yeah they never explicitly defined it
12:17:33 From Estefany Mendoza : Micro states are specific configurations of the system but I don’t think macro states were defined
12:18:07 From Danielle Thibeadeaux : Aren’t macro states just Eth and Eb?
12:18:18 From Estefany Mendoza : we also learned about different assumptions we should make
12:18:34 From Casey Davis : A "macrostate" (sometimes just called a "state") is a general big-picture description of the whole system, without getting into the specific details.
12:18:52 From Casey Davis : And a "microstate" is, yes, a more detailed description of the specifics.
12:19:11 From Casey Davis : What precisely they are depends on the system we're talking about.
12:19:35 From Estefany Mendoza : Thank you
12:19:36 From Noelle Tran : the coin example was understandable but once there are a large amount of macrostates it gets hard to keep track (like in the FNT)
12:19:56 From Casey Davis : So for coins, for example, let's say we're flipping 5 coins.
12:20:27 From Casey Davis : A microstate would be a specific listing of the heads/tails value of each coin (in order), such as HHTHT or HTHTT or TTTTT
12:20:42 From Casey Davis : In that case, by the way, how many possible microstates would there be?
12:20:50 From Danielle Thibeadeaux : 3
12:21:06 From Casey Davis : I listed three, but if we considered ALL possible microstates, how many?
12:21:29 From Jessica Marelli : 4
12:21:34 From Casey Davis : We could try listing them all out and counting, but that quickly becomes unfeasible for larger systems.
12:21:55 From Casey Davis : For instance if we just start listing
12:21:56 From Casey Davis : HHHHH
12:21:58 From Casey Davis : HHHHT
12:22:00 From Casey Davis : HHHTH
12:22:01 From Casey Davis : HHTHH
12:22:03 From Casey Davis : HTHHH
12:22:05 From Casey Davis : THHHH
12:22:09 From Casey Davis : HHHTT
12:22:10 From Casey Davis : HHTHH
12:22:13 From Casey Davis : and so on
12:22:23 From Casey Davis : The list gets very long, and longer still the more coins we're dealing with.
12:22:46 From Casey Davis : So instead of listing out all the microstates, we want a quick efficient accurate method for calculating how many possible microstates we can expect.
12:23:11 From Casey Davis : There's a whole branch of mathematics called "combinatorics" devoted to this sort of thing, the notion of rapidly counting up "number of ways that something could happen."
12:23:24 From Casey Davis : In this case, each coin has 2 options (heads or tails),and there are 5 coins.
12:23:37 From Casey Davis : What do we do with those five 2's to combine the possibilities?
12:23:55 From Jessica Marelli : 5 to the 2
12:24:07 From Casey Davis : Close, but in this case it's the other way around:
12:24:25 From Casey Davis : 2 options for the first coin, 2 options for the second coin, 2 for the third, 2 for the fourth, 2 for the fifth coin
12:24:33 From Jessica Marelli : ohh
12:24:35 From Casey Davis : Multiply all those twos together: 2•2•2•2•2
12:24:37 From Noelle Tran : 2^5
12:24:38 From Casey Davis : or 2^5
12:24:41 From Casey Davis : Exactly.
12:25:15 From Casey Davis : More generally, if you're dealing with several independent events ("independent" meaning the outcome of one event doesn't influence the other events), then you can multiply the individual microstates for each event to get the total number of microstates.
12:25:24 From Casey Davis : One coin has 2 microstates; N coins will have 2^N microstates.
12:25:42 From Casey Davis : We usually use a capital omega (Ω) to represent the number of microstates.
12:26:04 From Casey Davis : And the big assumption here is that all the possible microstates are EQUALLY LIKELY.
12:26:24 From Casey Davis : There's no reason to expect, for instance, that HHTHT will be any more or less likely that TTHTT
12:26:49 From Casey Davis : So we have 2^5, or 32, possible microstates. They're all equally likely. So what's the probability of each one?
12:27:55 From Casey Davis : In any situation, what should all the probabilites add up to?
12:28:03 From Estefany Mendoza : 1
12:28:04 From Kimia Tavassoli : 1??
12:28:29 From Casey Davis : Exactly.
12:28:50 From Casey Davis : If we're listing all possible results, the probabilities should add up to 1, or 100%, because there's a 100% chance that *something* is gonna happen.
12:29:01 From Casey Davis : So we have 32 equal pieces that have to add up to 1; what's the value of each piece?
12:29:06 From Shwetha Sekar : 1/32
12:29:08 From Casey Davis : Right!
12:29:33 From Casey Davis : So if there are Ω microstates total, then each microstate has a 1/Ω probability of happening.
12:29:47 From Casey Davis : All equally likely, because there's no reason any one microstate would be "favored" over the others.
12:30:05 From Casey Davis : Now let's take a look at "states" or "macrostates."
12:30:33 From Casey Davis : A macrostate would be a big-picture description of the whole system (all five coins) without getting into the specific details (the individual heads/tails value of each coin).
12:30:46 From Casey Davis : For instance, if I say "All the coins are heads," that's a macrostate.
12:31:20 From Casey Davis : Or if I say "There are more heads than tails," or "there is exactly one head" or "there are at least two heads," or "all the coins are showing the same side," those are all examples of macrostates.
12:31:45 From Casey Davis : And note that the macrostates *aren't* all the same probability. "At least two heads" is presumably much more likely than "all heads."
12:32:09 From Casey Davis : The key here is that each MACROstate is "associated with" some collection of MICROstates!
12:32:43 From Casey Davis : For example, if I described the macrostate "exactly one head," what would be some microstates that match that description?
12:33:16 From Danielle Thibeadeaux : HH or HT
12:34:05 From Estefany Mendoza : Would TH also be one?
12:34:17 From Casey Davis : Note that we're talking about 5 coins, though.
12:34:28 From Casey Davis : So we want a list of 5 values, exactly one of which is H.
12:34:33 From Shwetha Sekar : HTTTT
12:34:34 From Danielle Thibeadeaux : HTTTT
12:34:36 From Casey Davis : Right.
12:34:56 From Estefany Mendoza : THTTT
12:34:58 From Casey Davis : And something like THTTT would count as a *different* microstate that *also* matches the macrostate "exactly one head," because the five coins are distinct events.
12:35:00 From Casey Davis : Yeah.
12:35:10 From Casey Davis : So how many such microstates are there?
12:35:17 From Shwetha Sekar : 5?
12:35:18 From Danielle Thibeadeaux : 5
12:35:23 From Casey Davis : Yes, 5 exactly.
12:35:31 From Casey Davis : Because there are 5 places that that 1 head could go.
12:37:15 From Casey Davis : So if the macrostate "exactly one head" has 5 microstates associated with it, out of 32 microstates total, what's the probability of exactly one head?
12:37:36 From Shwetha Sekar : 5/32
12:37:39 From Casey Davis : Right.
12:37:45 From Casey Davis : And we'd presumably abbreviate that as
12:37:49 From Casey Davis : P(1H) = 5/32
12:38:01 From Casey Davis : because it's a huge pain to write out "probability of ___ heads" over and over again.
12:38:49 From Casey Davis : So probability is *additive* when we're talking about "or" combinations, in the sense that the macrostate "exactly one head" could be HTTTT *or* THTTT *or* TTHTT *or* TTTHT *or* TTTTH
12:39:05 From Casey Davis : "or" in this context means we add 1/32 + 1/32 + 1/32 + 1/32 + 1/32 to get 5/32
12:39:18 From Casey Davis : Since all those 1/32s are identical, we can just treat it as 5*1/32 or 5/32
12:39:58 From Casey Davis : "and," on the other hand, generally means multiplication. This is why we can multiply the 2's together in the first place: HTTTT, for instance, means the first coin is heads AND the second coin is tails AND the third coin is tails, etc.
12:40:14 From Casey Davis : so we multiply 1/2 • 1/2 • 1/2 • 1/2 • 1/2 = 1/32 as the probability for each *micro*state.
12:40:25 From Casey Davis : Any questions on that so far?
12:40:47 From Casey Davis : Or any other questions on the coin-toss example in general?
12:41:28 From Casey Davis : Also, some general tips for dealing with any coin-toss situation:
12:41:58 From Casey Davis : If you have a list of non-overlapping macrostates (such as "no heads," "one head," "two heads," "three heads," "four heads," "five heads"), the probabilities have to add up to 1.
12:42:30 From Casey Davis : The probability of "no heads" and the probability of "all heads" will always be 1/(number_of_coins), because there's only one way to get all heads and only one way to get no heads.
12:42:30 From Jessica Marelli : what would an overlapping macrostate me
12:42:35 From Jessica Marelli : be*?
12:42:54 From Casey Davis : Two macrostates "overlap" if there's at least one microstate that counts as associated with both.
12:43:17 From Casey Davis : For example, in the 5-coin situation, consider the macrostate "at least one head" and the macrostate "more heads than tails."
12:43:38 From Casey Davis : The microstate HHHTT is associated with both of those macrostates, because it has at least one head and it has more heads than tails.
12:44:18 From Casey Davis : It's generally easier to analyze a situation if we can "partition the probability space," that is, write out a list of macrostates that *don't* overlap, and that together cover all possible microstates.
12:44:31 From Casey Davis : That way you know the probabilities of all the macrostates in that list have to add up to 1.
12:45:29 From Casey Davis : For the five-coin example, if we consider the macrostates "no heads," "one head," "two heads," "three heads," "four heads," "five heads," then every MICROstate is associated with exactly one of those categories.
12:46:39 From Casey Davis : So let's try actually running the numbers on that 5-coin example.
12:46:54 From Casey Davis : Each macrostate probability should be a fraction of the form
12:47:12 From Casey Davis : (number of microstates associated with THIS macrostate) / (total number of microstates)
12:47:24 From Casey Davis : For 5 coins, how many microstates are there in total?
12:47:33 From Danielle Thibeadeaux : 32
12:47:47 From Casey Davis : Right. So each macrostate probability can be written as (something)/32.
12:48:18 From Casey Davis : Of course, a lot of those fractions might be reduceable, but it's often best to leave them in unreduced form—all with the same denominator—so it's easier to compare which ones are more or less likely than others.
12:48:47 From Casey Davis : Alternatively, converting all of them to decimal (or percentage) makes it easier to compare with probabilities in other situations that might be using a different denominator entirely.
12:49:12 From Casey Davis : So we already know that "no heads" and "all heads" have 1 microstate each (TTTTT and HHHHH, respectively).
12:49:21 From Casey Davis : That means P(0H) = 1/32 and P(5H) = 1/32
12:49:52 From Casey Davis : And we saw earlier that P(1H) = 5/32, because there are 5 places that the 1 head could go.
12:50:32 From Casey Davis : That still leaves us with P(2H), P(3H), and P(4H) left to calculate—the probability of getting 2 heads, 3 heads, or 4 heads.
12:50:40 From Casey Davis : Any thoughts on how we might calculate any of those/
12:50:42 From Casey Davis : ?
12:51:01 From Jessica Marelli : 2 to the 2
12:51:03 From Jessica Marelli : over 32
12:51:20 From Jessica Marelli : 3 to the 2 over 32
12:51:27 From Jessica Marelli : and 4 to the 2 over 32
12:51:27 From Casey Davis : They're all going to be *something* over 32… but to figure out the numerator, we want to count microstates.
12:51:45 From Casey Davis : I think 4 heads will probably be the easiest, because 4 heads also means how many tails?
12:51:52 From Jessica Marelli : 1
12:51:52 From Danielle Thibeadeaux : 1
12:52:08 From Jessica Marelli : so it would be 1/32
12:52:24 From Casey Davis : And there are 5 places to put that one tail, so the 4H macrostate should involve 5 microstates.
12:52:33 From Casey Davis : THHHH, HTHHH, HHTHH, HHHTH, and HHHHT
12:52:36 From Jessica Marelli : I am just confused how you would find like 5 are heads out of 100
12:52:48 From Casey Davis : So P(4H) = 5/32
12:53:36 From Casey Davis : because there are 5 microstates that match the description "exactly four heads."
12:53:50 From Jessica Marelli : is there a formula to figure this out or do we have to write down every possibility
12:54:46 From Casey Davis : For small systems, it's often faster to either write out the individual microstates or think through how many are leftover after counting the easier ones.
12:55:04 From Casey Davis : For larger systems, of course, that's usually not feasible, so we work out a formula instead.
12:55:34 From Casey Davis : In the 5-coin situation, we already have 1/32, 5/32, 5/32, and 1/32, so how much is left over to make it all add up to 1?
12:55:57 From Jessica Marelli : 20/32
12:56:01 From Casey Davis : Right.
12:56:28 From Casey Davis : And since there are only two macrostates left unaccounted for (2H and 3H), symmetry suggests that those 20 microstates are evenly split up: 10 each.
12:56:40 From Casey Davis : So P(2H)=10/32 and P(3H)=10/32 also.
12:57:28 From Casey Davis : Note that 2H and 3H (close to "half heads, half tails") are much more likely than 0H and 5H. Typically the results "near the middle" are much more likely than the results "on the edges," simply because they have far more possible microstates.
12:57:50 From Casey Davis : In fact that's really all probability is measuring: a macrostate is "more likely" if and only if it has more associated microstates.
12:58:17 From Casey Davis : Oh, and for larger collections of coins, the formula you're looking for is the "combination" formula, nCr.
12:58:44 From Casey Davis : nCr (often read as "n choose r") is a formula that tells you how many ways there are to choose "r" things if there are "n" things available.
12:59:07 From Casey Davis : So if there are 100 coins, and we want to know how many ways there are for 5 of them to be heads, we would want to calculate 100C5 (100 choose 5).
12:59:16 From Casey Davis : The actual formula uses factorials:
12:59:28 From Casey Davis : nCr = n! / ( r! • (n-r)! )
12:59:43 From Casey Davis : So 100C5 = 100! / ( 5! • 95! )
13:00:02 From Casey Davis : That would be the numerator for P(5H when flipping 100 coins)
13:00:07 From Casey Davis : And how would you find the denominator?
13:00:48 From Casey Davis : 100 coins, each could be H or T, so total number of microstates is Ω = ???
13:01:50 From Casey Davis : 2 options for the first coin, 2 options for the second coin, 2 options for the third coin, ……… , 2 options for the hundredth coin
13:01:59 From Danielle Thibeadeaux : 2^100?
13:02:02 From Casey Davis : Exactly.
13:02:16 From Casey Davis : So P(5H) = 100C5 / 2^100
13:02:38 From Casey Davis : The individual numbers involved get huge, but 100C(anything) is always less than 2^100, so the probability of course will still be less than 1.
13:02:59 From Casey Davis : And in general that nCr / 2^n formula should work for any number of coins.
13:03:23 From Casey Davis : For small situations, I'd recommend actually listing out the microstates so you can see how they're categorized—but of course that doesn't work so well for large systems.
13:04:09 From Casey Davis : And for the "scatter three blocks of different colors on a 10x10 grid" FNT question, what would a "microstate" be?
13:04:40 From Jessica Marelli : red blue and green
13:04:41 From Jessica Marelli : blocks
13:04:43 From Casey Davis : Yeah.
13:05:13 From Casey Davis : We want a "specific detailed desription of each part of the system," so in this case that means something like "red in space 37, blue in space 1, green in space 92"
13:05:30 From Casey Davis : In this case each "microstate" is essentially a list of three numbers in order, describing the exact location of each block.
13:05:39 From Casey Davis : Note that they could all be in the same space too.
13:06:11 From Casey Davis : Meanwhile a "macrostate" would be a more general description like "the red and green blocks are in the same space" or "all blocks are in different spaces" or "blue and green are adjacent to each other."
13:06:17 From Jessica Marelli : so a micro state is the color and location not just the three different colored blocks?
13:06:41 From Casey Davis : The thing is, the colors aren't changing from time to time. The locations are changing as the blocks move around.
13:06:49 From Casey Davis : So a microstate would be describing the *location* of each block.
13:07:33 From Casey Davis : Since the colors are all the same, we could just describe each microstate as an order triplet of numbers, like {2, 91, 27} with the understanding that, for instance, the first number is always the location of the red block, second is blue, third is green.
13:07:48 From Casey Davis : So mathematically each microstate could be written as three numbers in order.
13:08:03 From Casey Davis : Note for example that {91, 2, 27} would count as a different microstate.
13:08:25 From Casey Davis : So try applying what you know about microstates, macrostates, and probability to that blocks-on-a-grid situation.
13:08:53 From Casey Davis : I'm going to post the text of the chat on math.andcheese.org in case anyone needs to refer back to it.
13:09:05 From Casey Davis : See you next time!
13:09:06 From Danielle Thibeadeaux : Thank you!
13:09:09 From Casey Davis : You're welcome!
13:09:11 From Jessica Marelli : thanks!