[gwnn]

I must comment too that while I know that some of the stuff they teach in US universities we learned in high school in Romania, I know through experience that even many BSc Physics students including sometimes myself had problems recalling a lot of that knowledge and the professors needed to go through the material again. It was essentially a double time waste, first the high school teachers needed to teach us something which was over our heads and second the university professors needed some time to realise that we didn't know what we were supposed to. I hope a mod will move/delete this and related posts.

[kenberg]

 gwnn, on 2013-August-20, 08:04, said:

I hope a mod will move/delete this and related posts.

I second this. I just could not let it stand unaswered, but I would be most happy to get back on topic.

[Trinidad]

Please move this post along with the others.

It obviously depends on the school system. If you have a school system where at the age of 12, the brightest 10% of the kids are selected to go to "pre university high schools" and you let these kids chose at the age of 15 about 7 topics to focus on for the last 3 years of high school while they can drop all others, then it is not a miracle if those students who picked physics, chemistry, biology, math I (calculus and statistics) and math II (geometry/linear algebra) know a little bit more about mathematics than the average college freshman.

However, if you would take the bottom 10% and chose from those the ones who preferred cooking, brick laying, carpenting, etc. it will be hard to find one that can compete in math with the average college freshman.

Rik

[hrothgar]

 Zelandakh, on 2013-August-20, 05:10, said:

On the same seminar course was the problem of finding the expected win/loss for the game of craps - that was far more difficult (and interesting).

20 odd years ago, I taught a section of Introduction to Probability and Statistics at Indiana University.

I recall using the Birthday paradox during an early class...
However, early lectures are pretty much designed to make sure that everyone is up to speed and is familiar with the basics.

While I got some of my best teacher evaluations for that course, I didn't get a chance to repeat teach it...
Apparently too many of my examples involved blackjack and craps...

I tried explaining that Probability and Stats was originally developed to model games of chance and to this day the best examples involve gambling to no avail.

[billw55]

 Trinidad, on 2013-August-20, 08:33, said:

Please move this post along with the others.

It obviously depends on the school system. If you have a school system where at the age of 12, the brightest 10% of the kids are selected to go to "pre university high schools" and you let these kids chose at the age of 15 about 7 topics to focus on for the last 3 years of high school while they can drop all others, then it is not a miracle if those students who picked physics, chemistry, biology, math I (calculus and statistics) and math II (geometry/linear algebra) know a little bit more about mathematics than the average college freshman.

However, if you would take the bottom 10% and chose from those the ones who preferred cooking, brick laying, carpenting, etc. it will be hard to find one that can compete in math with the average college freshman.

Rik

Agree, a common fallacy of comparing of comparing exceptional examples from one (in this case geographical) subset to average examples from another subset. This does not prove much about the relative merits of the subsets.

Compare, for example, movies: comparing this month's offering at the local multiplex to Casablanca or Vertigo, and concluding that movies are much worse nowadays than in the past.

Or Kenberg's European passerby: was probably an academic herself, and hence probably not an average high school student.

[kenberg]

Teaching the "Probability for the masses" course I would tell students that I would be explaining how to compute some of the various probabilities associated with standard gambling games, but they could, in real life, approach it differently. Go to Las Vegas, look at the massive garish casinos, and ask where the money to build and operate them cames from. After that sinks in figure that unless you believe that you have a guardian angel who for some bizarre reason has taken it as his/her angelic purpose to help you beat the odds, maybe you should keep your credit card in your pocket.

I have more than a few criticisms of American education. Maybe move this to the WC? Not that it is all that easy to say anything original on the topic.

[gwnn]

BTW I still can't believe they taught us powers in 4th grade (~10 year olds). why???

[billw55]

 kenberg, on 2013-August-20, 08:54, said:

Teaching the "Probability for the masses" course I would tell students that I would be explaining how to compute some of the various probabilities associated with standard gambling games, but they could, in real life, approach it differently. Go to Las Vegas, look at the massive garish casinos, and ask where the money to build and operate them cames from. After that sinks in figure that unless you believe that you have a guardian angel who for some bizarre reason has taken it as his/her angelic purpose to help you beat the odds, maybe you should keep your credit card in your pocket.

The odds favoring the casino is certainly one reason they make money, but far from the only. I am convinced that even if they set the odds to slightly favor the player (51-52%), they would still make a profit because of loss lock-in. Most gamblers quit when they are behind (run out of money) but not when they are ahead. Hence losses are in the bank, but wins are negotiable.

[helene_t]

 billw55, on 2013-August-20, 09:04, said:

The odds favoring the casino is certainly one reason they make money, but far from the only. I am convinced that even if they set the odds to slightly favor the player (51-52%), they would still make a profit because of loss lock-in. Most gamblers quit when they are behind (run out of money) but not when they are ahead. Hence losses are in the bank, but wins are negotiable.

huh? If they set the odds to favour the player they would lose. It is as simple as that. Suppose they sell coin flips for $1 and pay out $2.01 cents for a head, after one million coins they would expect to have payed $1,005,000 for a net loss of $5000.

But some games, for example blackjack, have suboptimal strategies, so they can set the payout rate for the optimal strategy to something very close to 1 and still make a profit on all the suckers.

[billw55]

I meant it as an opinion of human gambler behavior, not a game theory computation. Many gamblers tend to bet until they are broke, then go home.

[helene_t]

sure. but that won't influence the profit of the casino.

[nate_m]

 kenberg, on 2013-August-20, 05:28, said:

Edit: OK, I now see how restricted choice plays a role in a spade lead from four cards. Holding 4=4=4=1 he might have led a heart instead of a spade, holding 4=3=4=2 he would lead a spade, except if the diamonds were something like KQJx.. So restricted choice says that the lead of a spade from a four card holding is evidence for a doubleton club. It's not just a matter of counting possible hands, we must also consider that a spade was led instead of a heart. The reason may well be that he has only three hearts.
This is in the "he is known to have four spades" situation, I started off looking only at the "known to have five spades" case.I still believe RC, or Monty Hall, or whatever you wish to call it, does not apply in the fivee card case.




Well,his title refers to Monty Hall, which is the classical example of restricted choice. My own history with this is: I was at a restaurant when a former student came up and gave me the Monty Hall problem This was back shortly after the Ask Marilyn article appeared in Parade. He explained it carefully, I thought for a bit, and said "The answer is 2/3, this is just like a problem in bridge where it is referred to as restricted choice"


A couple more points.

I had not gotten as far as Adam. And here is where restricted choice type arguments really have a role. If he led a four card suit, and if we can safely (for the purpose of the problem) infer from this that he has no five card suit, then indeed he is unlikely to have a stiff club.One can reason that he then might have led one of his other four card suits, or more simply (and preferably imo) simply reason that two clubs and 4-4-3 in the other suits can happen in more ways than one club and 4-4-4 in the other suits. This is because two clubs opens up the possibility of 4=4=3=2, or 4=3=4=2. And really we should just count leading from four spades as leading from a four card major.


It seems to me that one can (again for purposes of analysis) think of the situation as equivalent to the following: INstead of W leading, the game goes as follows. Before a card is led, dummy comes down. Declarer plays a club to the board and a club back, and sees three spot cards. Before he plays from hand, he gets to ask W "Do you hold at least one five card suit?" and W will answer truthfully. Basically, as I read the problem, that's what happens. At crunch time, declare knows the answer to this hypothetical question, and knows nothing else exacept that if declarer has a five card suit, one of those suits is spades.


Now again Adam observes that the choice between two five card suits is not random. Correct. Unlike the usual restricted choice situation, I think this situation is way too loaded with unlikely hypotheticals to be very useful.


One further thought occurs to me: With no information at all to go on, we play for the drop. But if knowledge that W has no five card suit makes it more likely that we should play for the drop, then knowledge that he has a five card suit makes it less likely we should play for the drop. The books have to balance here.The initial prbability of Qx in W's hand is the sum of the probability of Qx when holding five times the probability that he has at least one five card suit and the probability he has Qx when not holding five times the probability of no five card suit. If the probability of Qx when he has no five card suit is less than the initial probability of Qx, then the other has to be more.It takes more effort than i have expended to figure out how much more.
I realize I have phrased the above rather badly, but the point is that the books have to balance.

Bottom line: Play from the drop if the original lead was from four, finesse if it was from five. Which is pretty much what anyone would do, I think.

The inference about the lead from 4 card suits is even stronger than I think you give it credit for. If the opening lead was from 4, it is often correct to finesse opening leader for the queen, even with 9 cards between your hand/dummy. This is certainly true if there are 2 eight card fits for the defense, and I'd guess in the example given on BW hooking through opening leader is a small favorite in the event of a lead from 4.

[barmar]

 Zelandakh, on 2013-August-20, 05:10, said:

This is a university level problem in America?!

When I took Probability & Statistics at MIT around 1980, much of the introductory material overlapped with things I'd learned in high school math classes. But I can easily imagine that students who don't have a background that will get them into a school like MIT might not cover it as well in high school.

[ArtK78]

 Zelandakh, on 2013-August-20, 05:10, said:

This is a university level problem in America?! In England, I was given it to solve at a maths seminar somewhere around 13 or 14. I could see it as a first task in a "Basic Computer Science for Mathematicians" course perhaps but as a probability problem it is way too simple for university undergraduates. On the same seminar course was the problem of finding the expected win/loss for the game of craps - that was far more difficult (and interesting).

You have to start somewhere.

Courses in statistics and probability were not offered in our curriculum prior to college. I did have advanced calculus, however.

By the way, the odds of winning in craps was also one of the earliest problems in the course. If I remember correctly, playing pass results in a 49.47% win probability. And it is not a difficult problem.

[kenberg]

 nate_m, on 2013-August-20, 09:54, said:

The inference about the lead from 4 card suits is even stronger than I think you give it credit for. If the opening lead was from 4, it is often correct to finesse opening leader for the queen, even with 9 cards between your hand/dummy. This is certainly true if there are 2 eight card fits for the defense, and I'd guess in the example given on BW hooking through opening leader is a small favorite in the event of a lead from 4.

Ah yes, I see. And I tentatively agree. in fact, so he said on BW. I was still just too wrapped up in the five card issue to see this.More to think about.

[GreenMan]

 billw55, on 2013-August-20, 09:21, said:

I meant it as an opinion of human gambler behavior, not a game theory computation. Many gamblers tend to bet until they are broke, then go home.

But if you change the odds to favor the player, fewer gamblers will go broke.

[Stephen Tu]

 billw55, on 2013-August-20, 09:04, said:

The odds favoring the casino is certainly one reason they make money, but far from the only. I am convinced that even if they set the odds to slightly favor the player (51-52%), they would still make a profit because of loss lock-in. Most gamblers quit when they are behind (run out of money) but not when they are ahead. Hence losses are in the bank, but wins are negotiable.

This is very silly if you think about it more deeply. One has to quit sometime due to fatigue/hunger/etc. Because the games are rigged in favor of the house, most people lose when they play long gambling sessions, hence at quitting time you naturally find most of them are behind. If you in contrast rigged the games in favor of the player, the opposite would be true, after long sessions most people would be *ahead*, and most people would be quitting when they are ahead.

Now, it is possible for casino to run games that are slightly EV- against the house, if played perfectly. It's even been done (some video poker machines, blackjack with counting). That's because the vast majority of players *do not* play at all close to perfectly, and many play atrociously badly giving the house an extra 5-10%. A mass population of players playing badly (it's astonishing how people won't even bother to learn basic strategy at blackjack, doing things like never hitting on 16 regardless of upcard, etc.) can provide enough profit to cover a few perfect +EV players. But of course casino doesn't like to lose anything they don't have to, so +EV video poker is now nearly impossible to locate, and if they think you are card counting at 21 they will ask you to leave or play something else.

Quote

Many gamblers tend to bet until they are broke, then go home.

That's because they are playing -EV games and will nearly always run out of money given long enough session and bringing non-ridiculous sum of money to casino. I am a +EV poker player. If I go into a limit poker game adequately rolled, against a normal set of players (not a bunch of sharks equal/better than me) rotating into the table, I am *never* going broke. Still, I go home, because at some point I am about to keel over from fatigue, or the game breaks because everyone else leaves.

[jjbrr]

 ArtK78, on 2013-August-20, 10:13, said:

If I remember correctly, playing pass results in a 49.47% win probability.

If you define winning as, in the immortal words of chim17, "being pretty drunk, betting on 11, yelling 'Yoleven!' and hitting it" sometimes, I'm pretty sure win probability is 100%.

[GreenMan]

 ArtK78, on 2013-August-20, 10:13, said:

By the way, the odds of winning in craps was also one of the earliest problems in the course. If I remember correctly, playing pass results in a 49.47% win probability. And it is not a difficult problem.

I don't have the numbers handy, but I heard a while back that Don't Pass is the better bet by a small margin, though it's still -EV because the house keeps a small cut. In fact, unless you're a +EV player in some games as Stephen described, Don't Pass is the best bet in most casinos.

[gnasher]

 kenberg, on 2013-August-20, 05:28, said:

Well,his title refers to Monty Hall, which is the classical example of restricted choice.

I tend to assume mathematicians are right about most things, but not necessarily about the format of a 1960s game-show.

As I understand it, what mathematicians call the "Monty Hall Problem" isn't the same as what Philip Martin calls the "Monty Hall Trap" - in former you're offered a chance to switch doors; in the latter you're offered the chance to swap your chosen door for some cash. I've no idea whether both, one or neither of these scenarios actually occurred on Monty Hall's game show. Do you know? If you don't, the most you can criticise Philip Martin for is using different terminology from you.

Martin's main point isn't about restricted choice, which he mentions only in passing. His argument is about how to treat information that merely tells you something you already knew. Monty Hall opens a door and shows us a goat, but we already assumed that one door had a goat behind it, so he's told us nothing at all. LHO tells us that his longest suit is of 5 cards, but we already assumed that he had a 4.5-card suit, so he's only told us about half a card.