Probability Primer (for Scuba Divers)

[lloyd_borrett]

How about this:
You are lost in a cave with only one possible entrance/exit. While searching for the exit, you come to three possible exits, each appears equally likely. You pick potential exit "C" for no particular reason other than intuition. A diver emerges from potential exit "B" and signals that "B" is not an exit. What should you do? Why?

Actually this isn't quite an accurate representation of the Monty Hall paradox because it isn't stated that the diver signalling that exit "B" isn't an exit knows where the real exit is, knows what choice you've already made, and is now giving you a chance to switch your choice to exit "A" or stay with your choice of exit "C".

If the diver is just someone who doesn't know which of the three options is the real exit, who simply got to the junction first and tried exit "B" before you, then your chances of it being exit "C" or exit "A" remain an overall 1 in 3 proposition. It's only when the whole scenario is in place that the Monty Hall paradox applies.

Best regards, Lloyd Borrett.

 

[terrylowe]

I find that most people either cannot or will not attempt to understand probability. They respond more to what has been told to them through the media and through their brother-in-law's mother's second cousin's son.

For example: In my medical practice >>
"Your child has the flu Mrs Jones" (average #deaths in the USA yearly is 31,000)
"My God doctor! I mean he really IS sick! Aren't you going to test him for West Nile???!!!!!" (average #deaths in the USA yearly is <300)

In another example:
"You need a flu shot Mrs Jones" (In the 2009 global H1N1 pandemic, estimates of flu deaths are from 300,000 to 500,000)
"Doctor, isn't a flu shot DANGEROUS?" (no statistics available regarding the death rate from taking flu shots because there is obviously a governmental coverup involving this topic)

Based on what the media is saying (and many folks here on SB are implying), diving is excessively dangerous and you will die from doing it if you do it long enough. I mean, who in their right mind would dive in the same water as sharks??????

Point is, although statistics have some incredible real world applications, real people often cannot or will not apply statistics to what they are doing (Casinos bank on this)

 

[vladimir]

.....

 

[lowviz]

Actually this isn't quite an accurate representation of the Monty Hall paradox ...//...

Yes, you are correct. However, it was set up to work exactly the same way while being stated as simply as possible.

The whole thing hinges on new information being given after the initial choice. In the Monty Hall problem, Monty is an informed helper. You pick a door then he shows you where the prize isn&#8217;t, down to one last door. Now you have two doors and most people just can't let go of assuming that there is 50/50 chance of the prize being behind either door. Wrong.

Let&#8217;s say that there are a million doors. You randomly pick one. Monty opens all but one of the remaining doors. Would you change your pick?

 

[Akimbo]

&#8230;The real power gained by being armed with knowledge of the basic rules of probability (besides being able to craft a cogent probability argument or decipher a flawed one), I think, comes from being able to try different probabilities&#8212;different actual numbers&#8212;no matter what particular method you use for assigning these probabilities. (You must be careful, though, to assign these probabilities coherently!) A lot of insight can be gained sometimes simply by trying different probabilities&#8230;

I completely agree and thank you for this post. In my limited experience, this methodology is only applicable to emotionally detached analysis like to certifying agencies, insurance wonks, and management that needs CYA.

Make it personal (as in my butt on the line), and the equation usually gets biased toward the highly improbable and overlooks the very real dangers that complexity adds to rapid and accurate diagnosis of problem(s). It does not matter if that regulator in your mouth has a tenth of the failure probability as your front tire. For some reason we are more frightened of suffocation than crashing into a bridge pillar.

Forget the math if you can&#8217;t relate to it. What matters is going through the what-if analysis until you are satisfied. The never-ending process will make you a better diver.

 

[lowviz]

...//...Forget the math if you can’t relate to it. What matters is going through the what-if analysis until you are satisfied. The never-ending process will make you a better diver.

That is the whole idea...

Philosophy | BetterExplained

Quoting from the above link:

"Math is no more about equations than poetry is about spelling. Equations and spelling exist to convey an idea. Understand that idea."

 

[Diogenesdemar]

I'm still waiting to hear if "4" was the correct answer . . .

 

[rx7diver]

I have a degree in mathematics, so let me, for a moment, nurse a pet peeve. The original post is typical of what often passes for explanation in the field. If you have a little bit of math training it is perfectly clear, but you have no use for it. If you don't have any math training it is all but incomprehensible, and leaves you thinking that the author is very smart. I'm sure you could compose a legal brief that would be similarly incomprehensible to the uninitiated, and similarly useless in helping him understand the issue in question.

Vladimir,

What I presented in my original post is basically a quick summary of some of what is taught here in the US in a high school probability and statistics course (generally a high school algebra-based course)--same concepts, terminology, and formulae, but necessarily without the "value added" that a master teacher would bring.

People even just half my age will have been away from high school for a number of years and easily might have forgotten this stuff--if they had taken the course and learned this stuff at all. Others might never have learned this stuff. Anyone who employs and/or seeks to digest a probability argument "needs" to know this stuff, imho, at the very least. I hope my primer will be an aid to anyone who wants/needs it.

The power of the internet is, it facilitates learning from each other. (I personally have learned a great deal from people here on SB who have in-depth knowledge in fields and disciplines I personally know little about.) I sincerely hope people here will correct anything I've written that requires correcting, fill in any omissions that I've made that they think need to be included, and clean up my presentation to make it less "incomprehensible" if they think it needs to be made so.

Safe Diving,

rx7diver

 

[vladimir]

Kudos for making the effort, RX7.

For anybody who is interested, Math Goodies offers interactive lessons in this subject (among others) that are easily understood by anybody.

 

[lsorenson]

Very cool if you can assign meaningful failure rates. You can't.

How about this:
You are lost in a cave with only one possible entrance/exit. While searching for the exit, you come to three possible exits, each appears equally likely. You pick potential exit "C" for no particular reason other than intuition. A diver emerges from potential exit "B" and signals that "B" is not an exit. What should you do? Why?

Working off the assumption that both are lost...

Scenario 1) I would make my peace with the boy upstairs and then haunt my Cave instructor for the rest of his life for not teaching me proper caving technique, assuming the second diver is lost as well...!!!

Scenario 2) Flip a coin to see who chases down A while the other chases C... Depending on distance and air consumption; maybe one of us would make it...

Scenario 3) Kill the second diver and take his air and hope I find the exit on your first guess and hope for enough air if I'm wrong...! (kidding here guys so lighten up; please

Oh, and to the odds of boys and girls... My father in-law was 0 for 5 with boys... all girls; 5 girls!!! Its a wonder he never went crazy then again, maybe he did and hid it well...!

All in fun, lee