Explain to me this probability puzzle

[Piratecat]

Navior, thanks for the clearest explanation I've seen of this phenomena. Statistics are cool.

 

[Pielorinho]

Here's the explanation that finally convinced me. In order to explain it to you, I'm going to have to ask two favors.

First, understand that I'm neutral: like Monte, no matter what you choose, I'm going to offer you another choice -- even if you choose right the first time.

Second, I'm thinking of a number between one and a hundred, and I need someone to try to guess that number.

As soon as someone makes a guess, I'll continue with the explanation.

Daniel

 

[Umbran]

Okay, I'll bite. 23.

 

[Pielorinho]

Thanks, Umbran! As promised, like Monte, I'll give you another choice; but like Monte, I'm going to elimate all the other choices but one.

I'll tell you now that the correct number isn't between 1 and 22, isn't between 24 and 45, and isn't between 47 and 100.

Would you like to switch to another number, or are you happy with your first guess?

Daniel

 

[Navior]

Navior, thanks for the clearest explanation I've seen of this phenomena. Statistics are cool.

You're welcome!

 

[diaglo]

...As promised, like Monte, I'll give...
Daniel

i always knew you were a Monty Hauler...

 

[Pielorinho]

Total Monty Hauler here.

Anyone? Anyone? Given the information above, does anyone want to tell me what their guess for the number would be? For extra credit, does anyone want to tell me what the odds are that their guess is correct?

Daniel

 

[jgbrowning]

Total Monty Hauler here.

Anyone? Anyone? Given the information above, does anyone want to tell me what their guess for the number would be? For extra credit, does anyone want to tell me what the odds are that their guess is correct?

Daniel

I'd change to 46 and I think my odds would be 99% correct. *cross fingers*

joe b.

 

[Pielorinho]

Exactly right, jgbrowning!

Here's what I did: of all the choices, I let you choose one of them. Then I bundled all the other choices together. If you chose correctly originally, I selected one of the bundled choices at random to represent the bundle; if you chose incorrectly originally, I select the correct number from the bundle to represent the bundle.

99% of the time, you chose incorrectly originally; that means 99% of the time, I'll be choosing the correct number from the bundle to represent it, and 99% of the time, you'll benefit from switching.

1% of the time, you chose correctly originally; that means 1% of the time, I'll be choosing an incorrect number from the bundle to represent it, and 1% of the time, you'll lose by switching.

That's exactly what Monty Haul does in the "Let's Make a Deal" game, except instead of working with 100 choices, he's working with 3 choices.

And that's the explanation that finally convinced me that this statistical trick worked as advertised.

Daniel

 

[Jeff Wilder]

My friend Cassandra presents it in the Three Door format the first time, and then, after she reveals the answer and the subject starts arguing, she puts it this way:

"Okay, instead of three doors, imagine there are a million doors. You pick one, and Monty opens all of the others except one. Now do you see that you should switch?"

Basically, the idea that by opening one door -- leaving only one unopened -- Monty is providing a lot of information is lost on the listener, whereas when he opens 999,998 others -- leaving one unopened -- people get that he's helping out.