More Blogger News, and Bayes Theorem
Well I am pleased as punch today to have some more news of a biggest-ever tournament cash by one of our very own. The amazing thing is it is someone who I've already posted about a biggest-ever cash recently right here on the blog. But on Tuesday afternoon / evening, it happened again. This blogger stayed home from work after a redeye flight back from a Vegas trip of all things, played in the $165 buyin 32k guaranteed nlh tournament at 3pm ET on full tilt, and ended up losing a race to bust in third place. For over $5300. Cold. Hard. Cash. Who is this blogger dominator, you ask? Why, it is LJ. Again. And she's already got a post up about her latest run. That is two huge tournament scores for LJ within days, and three big ones in the past few weeks. How far will her tournament prowess extend one day? We'll all have to watch and see. Now go congratulate LJ, and if you can figure out what her new-found secret is, somebody please tell me.
So today I wanted to chat briefly about Bayes Theorem. There are actually complex mathematical formulas involved in this thing, which I accept as true without taking the effort to go through the detailed mathematics used in various conditional probability equations. But I do understand the effect of Bayes Theorem, including its tremendous applicability to the poker world, and to me it remains one of the most interesting, and truly difficult-to-grasp, theorems out there. And yes I know that "theorems" is probably not a word. See, I know it's not right, but I still ain't changing it. Deal with it.
So to introduce my discussion on Bayes Theorem this week, I'm going to start today with a couple of problems, and I would like as many of you as feel comfortable to provide your estimates or answers to the two questions I pose below in the comments to this post. Then tomorrow I will do my best to explain conceptually how I like to use the concept involved in Bayes Theorem to consider the two solutions.
Question #1: You are on the popular 70s game show Let's Make a Deal. Monty Hall is up there and he shows you 3 boxes on the stage, and asks you to pick one at random, explaining that under one box is a million dollars cash, but under the other two is a dead rodent of some kind. You choose Box #1. He then opens up Box #3 and shows you that there is in fact a dead rodent under that box. Now he offers you to either keep Box #1 or switch to Box #2, your choice. What do you do? Do you switch, do you keep your first box, or does it make any difference?
Question #2 (this one comes directly from The Mathematics of Poker): Suppose doctors have a screening process for, say, Lupus, and that if a person is screened who actually has Lupus, the screening process will return a positive result for Lupus 80% of the time. Assume also that if a person who does not have Lupus is screened, the test will return a (false) positive result for Lupus 10% of the time. Lastly, assume that we know that 5% of the total population on average actually has Lupus.
A person is selected at random from the population at large and screened, and the test returns a positive result for Lupus. What is the likelihood that this person, who just tested positive for Lupus, actually has Lupus?
Let me know your answers (either of these questions may be just estimates), and I look forward to discussing these two problems in more detail on Thursday. Then I will write some as well about how Bayes Theorem applies to poker and the decisions that you are faced with at the table.
I will try to register for the Mookie tonight at 10pm ET (password is "vegas1" as always) to chase my third consecutive cash in this event, but I will be at the coveted Yankees - Red Sox game this evening so I will probably be late to my one elusive blonkament that I can never win. Free blinds for everyone!
Labels: Bayes Theorem, Blogger Tournament Scores, Let's Make a Deal
posted by Hammer Player a.k.a Hoyazo @ 7:43 PM
32 Comments:
I don't know the answer to the Lupus questions, but you should always switch in the Monty Hall problem.
There's a cool little javascript app to illustrate the problem at http://mozy.com/blog/2006/09/11/probability-is-fun
You can run it by yourself or let it run on auto and can see how the odds lay out.
answer 1) Let's make a deal. I wouldn't switch. Even though door number 3 has a rat, and the other door that I didn't pick now has a 50% chance of having a million in it, my door now also has a 50% chance of having a million in it. It makes no difference that I was shown another door with a rat in it, other than the fact that my chances of winning just went from 33% to 50%. I could switch, I could stay the same, but either way I still have a 50% chance of winning.
answer 2) I say there's a 5% chance of the person actually having lupus. If I were that person, I would want to be tested about ten more times and see what the results were. If my overall results were 8 positive and 2 negative, I'd assume I had Lupus. If my results were 1 positive and 9 negative I'd assume I didn't have it. (I'd say if my results were within about 2 either way of those numbers I'd assume the same). But those percentages are so bad that I'd just stick with my answer of 5%, based on the 5% of the population having lupus and ignore the test result.
In the Monty Hall problem, assuming Monty has to open a box and show a dead rodent, then switching doubles your EV.
In the Lupus problem, 4/13.5 = 29.6% (total % of time a person with lupus is selected and tests positive => 0.05*.8 = 4%; total % of time test comes out positive => 0.95*0.1 + 0.05*0.8 = 13.5%)
Always switch. I'm betting the comments I haven't read have already explained why, but the basic deal is when you pick box #1, there is an approx. 33% shot at a million bucks.
So you're holding 33%. The 'field' is holding 66%. When Monty lifts box #3 to show the dead rat, the 'field' is narrowed to box #2. Which still holds 66% equity. The 'field's' equity has not shrunk with your new information. You just happen to know which of the two boxes now holds all of the 66%.
This was a Marylin Vos Savant question that got college mathematicians writing her to insist she was wrong. She was right.
The odds of somebody who is given a positive Lupus test actually having Lupus is 90%.
The rate of false positive is 10%, so this cat is basically a five outer with one coming.
The rate for false negative is irrelavent, since we aren't dealing with a negative result. So is the population percentage with Lupus.
thanks hoy. i can say w/ certainty that my recent cashes have nothing to do with my math prowess.
i would have said not to switch, however i trust that people far smarter than me have established that switching is the proper course of action.
as for question 2 -- ha! my head was spinning just thinking about the percentages. i'm having nasty flashbacks to the sat.
For the Lets Make A deal question it depends on what Monte's strategy is. If he is showing you a random box (could contain the money), then you switch. If he is showing you one that has a rat on purpose, then switching does not help. Since you do not know what his strategy is, you might as well switch.
For the lupus question, there is a 4% chance that you are positive and test positive (.05)*(.8). There is a 9.5% chance that you are negative and test positive (.95)(.1). So you have a 70% chance of a false positive, and a 30% chance of a correct positive.
In the first one, I don't understand how it could be different by switching. I mean we still don't know where the money is so if we assume that we already picked the right box, Monty could have opened either box and it wouldn't matter right? Plus, don't they always say, don't second guess yourself?
In the second question, if I am calculating things correctly, I think the likelihood of a positive result indicating someone actually has Lupus is pretty low. Actually, I didn't see it till now but on my ten-key (if you're a CPA you know what that is), the numbers that blinders mentioned were exactly what I punched in. So I agree with blinders. I came to something like 29.6%. My math could be way off but if it is, I'm blaming the calculator.
1. Go with your initial instincts. Don't 2nd guess yourself. Keep the original box.
2. There is a 95% chance of an 10% positive result and a 5% chance of an 80% positive result. So I would say it is a 13.5% chance that they test positive.
The person just tested positive, so there is 9.5 times out of 13.5 times it is false positive or 70.3% chance of a false positive = 30% chance that they are positive.
Since Monty shows you a rat no matter what you chose and you still have to choose between two options you are 50/50 and Bayes Theorom doesn't really apply. It's like looking at roulette results and thinking that since it's come 7 reds in a row it should now be black on the next spin.
As for the Lupus, if you are a follower of Occam's Razor then you should start saying your goodbyes.
I would probably get a second opinion.
Yeah, I came up with the same results as d for #2.
As far as #1 goes, I guess it doesn't matter. It's either in box #1 or it isn't. Choosing the other box won't give you better odds.
Question 1: I wouldnt switch.
I don't think it matters.
question 2: My head hurts thinking about it.
Monty Hall is such a great question because it's so counter-intuitive.
Two hints:
What are the chances of guessing right out of 3 boxes?
Since Monty can ALWAYS show you a wrong box, does him showing you anything change your original odds?
Question 2 person has 4/13.5 chance of having Lupus after testing positive.
Question 1 switch.
Ugh. Math. I say go with your initial instincts regardless of the outside information presented to you by Monty. I don't see how it makes any difference given that you're at 50 percent. That's how I could see it related to poker, how we sometimes let our minds outthink ourselves instead of going with our gut instinct.
I have no fucking clue on the second problem and that's why I am a writer.
#1: Always switch, it's +EV
#2: What d said.
Gotta switch boxes. You lose a lot of EV going with your gut rather than the math.
90% chance. Get some treatment and hope you were one of the 10% with a false positive.
Switch - IF this is the Marilyn Vos Savant example.
The switch answer is based on the Monty knowing which door has the prize, and ALWAYS picking a door that doesn't. You're missing that tidbit... if this isn't the case, then it's actually not the Vos Savant question, and it depends on your assumption on the probability of him picking that particular door based on where the prize is.
Since this will be related to poker, always picking where the prize isn't doesn't seem to apply.
As for Lupus - about 29.6%
(5%x80%)/(5%*80% + 95%*10%)
or
4/13.5 (# of lupus tested positive divided by total number of positives per 100)
This should be fairly instinctual since 10% of 95% > 80% of 5%, so there will be more false positives than true positives. That changes if the percentage of the population that has lupus goes beyond 11.875%. Just saying.
1) Switch. This is Monty. If he had selected to one with the cash, then the sad musinc would play and the game would be over. So whether he chose to show you the rat or chose to give you a chance to switch if a rat showed in his random choice does not matter. You still have information now that you did not before and that affects you choice.
2) 30% you have Lupus.
I'm wrong.
Not about Monte. the other thing. I just read a damn book with a similar example too.
I'll be interested in seeing how we're wrong, pirate. I assume we are since you've seen this in a book.
The thing about the lupus is that we ALREADY have a false positive. So there are two options:
1) He does have lupus. If he does, then he is ALREADY part of the 80% who have it and correctly get a positive result. The chances of him being among the 20% who have it and get a false negative are 0%.
2) He doesn't have lupus, but got a positive anyway. The odds of this have been established at 10%.
The fact that 5% of the population has Lupus is utterly beside the point. The fact that 20% of those who have it get negative results is utterly beside the point. Why? Because he already got a positive result. Either he has it (90%) or he doesn't have it (10%).
Second paragraph of the last post should read:
The thing about the lupus is that we ALREADY have a positive. So there are two options:
and not:
The thing about the lupus is that we ALREADY have a false positive. So there are two options:
Goat - the thing is, you have a better chance of having a FALSE positive than true one. Since you know you have a positive, you are now figuring out what the chances of it being a real positive are. 29.6%.
The problem lies in that it's so rare for someone to have lupus (in the example). Even if the test was 99% accurate and only had a 1% false positive rate, the fact that only 5% of the population has lupus means that a positive test would mean you had an 83.9% chance of actually having it. Hence the need for follow-up testing.
as a long time lurker of this blog i have to comment on the first question.
By staying with your decision you have a 33% chance of winning. By changing you have a 66% chance of winning.
There are people that will argue the world that at the time of switching you have a 50/50 chance but this isnt true.
i was about to explain the theory but noticed julius_goat has done a great job. As mentioned the collective other 2 boxes hold 66% of being right. you hold 33%. by knocking out 1 box from the collective the remaining still holds 66% which is now in 1 box.
this is easy to program in excel. If anyone wants me to then let me know. or run it yourself with 3 boxes and a peanut. by staying with your decision you win 1/3 by switching you win 2/3
(Darrenb on Full Tilt)
My head exploded shorty after I read Theorem. I should know this because I work on large cancer studies. Since I'm a systems person not a project person, I'll ask the MD who heads up our larger study what he thinks. Hope I get an answer before Hoy's next post as my source is in meetings all day.
1. Always switch. You double your chances of picking right.
2. 29.6%.
Just in case anyone gives a shit, I posted about Bayes' Theorem and poker here: http://gollyg.blogspot.com/2007/07/star-bright.html
Goat,
I actually tried to do the math on this one last night in between hands at a live table.
In a population of 1000, it is presumed 50 have lupus.
Of those 50, only 40 will test positive.
950 of the sample do not have lupus.
95 will test incorrectly as having lupus.
135 out of 1000 have tested positive and only 40 have lupus.
It appears your chances of actually having lupus are 40/135. Almost 30%.
I figured something was up when I realized so many people had false positives that it didn't make sense that all would have a 90% chance of having lupus.
I may still be wrong.
To everyone: you might be interest in a book Fooled by Randomness. It's eye opening in a lot of ways.
Yep, I started at the wrong place.
Bad goat. No tin can.
I understand what everyone is saying about how you should switch, but I think that you are missing one important detail here, and please, correct me if I'm wrong on this: You were never going to choose between three boxes, just two. The rub lies in the fact that Monty knows where the rats are and will ALWAYS show you one. The Savant outcome only works for choosing to reveal a random box.
In essence, the first choice was completely irrelevant, you could just skip it, since its predestined anyway.
I swear I haven't read any of the other comments... (if that would even matter)...
Starting with question 2, I believe the answer is simply 13.5%. That's, (95%*10%)+(5%*80%)= 13.5%. That seemed pretty basic.
Question 1: I immediately thought the answer was simply NO it does not matter either way! But, after posing the question to my roommate, he made me think a little. His immediate answer was switch to box 2. Explaining that when #3 was revealed you had technically already 'won' the first round with your initial choice, and are now being given the opportunity to make a different choice for the 50% chance. Going further to state that when one's first 2 out of 3 shot was a winner, why wouldn't they assume from there that that pick wasn't going to win twice in a row, and go the other way for the 1 out of 2 chance. (?!) But now I've now gone cross eyed thinking about it. And I think my roommate's drunk. Still think it doesn't matter. 50% chance is 50%... is it not? Looking forward to the next post, that explains the reason for the questions.
hoy,
i am not a blogger just play poker everyday when im not working. I read your blog everyday and really enjoy all that you write. And your tournament always sounds like a good time so i thought i would give it a go. it was a blast and filled with fun donks, so i will be back when im not working sincing i run a bar.
worm
1) Always switch, this has been proved so many times i am not going to reproduce it here.
2) The answer is ~7.7%
Shortest explanation is using Bayes' Theorem:
With H being the hypothesis - has Lupus, E the event of a +ve test, which is ~10.4% (9.6% are false positives + .8% true positives)
we have P(H|E) = P(E|H)P(H)/p(E)
= 0.8 x 0.01 / .104
= .077
= 7.7%
Edit to above, I used numbers from a similar prob oops! should have read closer before jumping in with "I know, I know!" ;-)
P(H) = 0.05
P(E) = 0.1 + (0.8 * 0.05) = 0.14
P(E|H) = 0.8
so P(H|E) = 0.8 * 0.05 / 0.14
= 28.5% Chance of Lupus given a +ve test result.
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