Millionaire, should he switch?

mikeymouse

Watching 'Who wants to be a millionaire' last night got me thinking;
The guy was on 125k question , he had 2 lifelines; 50/50 and ask the audience.(ignore ask the audience for this)
He had no idea as to the correct answer , but favoured A for no logical reason,
likewise he had no logical reason to discard B C or D.
So he had a 1 in 4 chance of A being right.
He took the 50/50 lifeline and was left with A and B.
So logically that's a 50/50 chance for both A and B,right?
Well,no.
It's not quite Monty Hall , but what are his chances now were he to switch?
(Bear in mind whenever he picks an incorrect answer the computer will discard it
2 out of 3 times)

Pac1Man

mikeymouse wrote: »

Watching 'Who wants to be a millionaire' last night got me thinking;
The guy was on 125k question , he had 2 lifelines; 50/50 and ask the audience.(ignore ask the audience for this)
He had no idea as to the correct answer , but favoured A for no logical reason,
likewise he had no logical reason to discard B C or D.
So he had a 1 in 4 chance of A being right.
He took the 50/50 lifeline and was left with A and B.
So logically that's a 50/50 chance for both A and B,right?
Well,no.
It's not quite Monty Hall , but what are his chances now were he to switch?
(Bear in mind whenever he picks an incorrect answer the computer will discard it
2 out of 3 times)

Variable change. Switching to B would increase his odds, my only worry being that the 50/50 removal process is not random and someone choose to allow the 'A' to remain based on his initial thoughts.

ardinn

A stands at a 25% of being right as he picked it at 1 in 4
B now stands at 75%

He should switch

facehugger99

ardinn wrote: »

A stands at a 25% of being right as he picked it at 1 in 4
B now stands at 75%

He should switch

That's not true if the computer doesn't know that he already has 'picked' A - which I assume is meant to be the case.

ardinn

facehugger99 wrote: »

That's not true if the computer doesn't know that he already has 'picked' A - which I assume is meant to be the case.

iv no interest in the computer - he picked a, a is still there.

facehugger99

ardinn wrote: »

iv no interest in the computer - he picked a, a is still there.

It doesn't matter what interest you have in it - it's the only important part of the scenario that affects the odds.

dubrov

Monty Hall doesn't apply here as the 50/50 was not guaranteed to remove answers not selected by the contestant. If it was, a switch would be preferable.

So the odds are 50/50 for the remaining options

abff

mikeymouse wrote: »

Watching 'Who wants to be a millionaire' last night got me thinking;
The guy was on 125k question , he had 2 lifelines; 50/50 and ask the audience.(ignore ask the audience for this)
He had no idea as to the correct answer , but favoured A for no logical reason,
likewise he had no logical reason to discard B C or D.
So he had a 1 in 4 chance of A being right.
He took the 50/50 lifeline and was left with A and B.
So logically that's a 50/50 chance for both A and B,right?
Well,no.
It's not quite Monty Hall , but what are his chances now were he to switch?
(Bear in mind whenever he picks an incorrect answer the computer will discard it
2 out of 3 times)

The decision would depend on whether or not the 50:50 choices are predetermined and purely random.

I don't see how the 50:50 choices could be manipulated to ensure that the preference expressed by the contestant remains as one of the options. This would imply that there is someone working behind the scenes who would decide what to present as the options after the contestant has said "I think it's A, but I'll go 50:50 just in case."

Apart from the fact that this would effectively be cheating by the company running the program, it would leave them wide open to manipulation by a contestant who's fairly sure it's not A, but doesn't know which of the other three it might be.

So if we ignore the possibility that the options presented would be influenced by an opinion expressed by the contestant, that brings us to the second point. How random are the 50:50 options?

I've watched the program over the years and it's not uncommon to have a question where two of the answers seem to be a lot more plausible/likely than the other two answers. If the 50:50 options were purely random, there would be a 1 in 6 chance of both these options remaining under the 50:50. There would also be a 1 in 6 chance that neither of these options would appear under the 50:50.

In either of the above cases, the contestant is not really helped by the 50:50 and it's more or less a toss up which one to choose from the two remaining options. In the other 4 out of 6 cases, the contestant is helped by the 50:50, assuming that the contestant's reasoning as to which are the most likely options is valid.

However, in practice, I've found that where there are two 'obvious' answers, they both remain in place after the 50:50 a lot more than 1 time out of 6. I think the difference is statistically significant (although I have to confess that I'm basing this on general observation rather than detailed analysis) and this suggests to me that the 50:50 is not completely random and may not improve the odds as much as the contestant might hope.

Bottom line is that the odds are definitely no worse than 50:50 after you have eliminated two of the wrong answers and I would see no reason for the contestant to change from their original answer if it remains as an option after they have used the 50:50.

ardinn

dubrov wrote: »

Monty Hall doesn't apply here as the 50/50 was not guaranteed to remove answers not selected by the contestant. If it was, a switch would be preferable.

So the odds are 50/50 for the remaining options

wasnt guaranteed - but as it transpired, it happened - is mh conditional that it all has to be pre arranged? no! Ill stick with my original assuption

dubrov

ardinn wrote:

wasnt guaranteed - but as it transpired, it happened - is mh conditional that it all has to be pre arranged? no! Ill stick with my original assuption

Monty Hall works because the host introduces new information about the unselected answers. The host never removes the correct answer so effectively increases the odds of the remaining unelected answers being correct.

In this case, the host could also have removed the selected answer so hasn't added any extra information about the unsetected answers

mikeymouse

dubrov wrote: »

Monty Hall doesn't apply here as the 50/50 was not guaranteed to remove answers not selected by the contestant. If it was, a switch would be preferable.

So the odds are 50/50 for the remaining options

The computer is 'supposed' to remove 2 wrong answers,regardless of what he picked, so any wrong answer he picks has a 66.% chance of being removed

Peregrinus

This is different from Monty Hall because in Monty Hall your provisional choice influences the subsequent elimination - you provisionally pick one option, and then the elimination is made only among the options that you haven't picked.

In Millionaire, this isn't the case. If the players is leaning towards A, this does not affect the likelihood that A will be eliminated. Suppose the correct answer is B; when the player opts for 50/50 the possible eliminations are AC, AD and CD. If the correct answer is A, the possible eliminations are BC, BD and CD. CD has the same likelihood in each case, so knowing that CD was is fact the elimination made gives no clue as to which of the two cases (A is correct; B is correct) in fact prevails.

The information given her that the player is leaning towards A is a giant red herring, since (unlike in Monty Hall) it hasn't in any way affected the eliminations that are possible, or the elimination that is in fact made. This means that while the elimination does give new information (C is wrong; D is wrong) it does not give new information as to the relative likelihood of the remaining options.

Before the elimination the chance that the correct answer was A was 25%, and the chance that the correct answer was B was 25%. After the elimination the chance that it is A are 50% and the chance that it is B is also 50%. This doesn't suggest that the player will improve his chances by switching to B.

mikeymouse

ardinn wrote: »

A stands at a 25% of being right as he picked it at 1 in 4
B now stands at 75%

He should switch

I agree with your 75% every time he is given the choice to switch,which answers the question.
but he is only given this choice about half the time ,
the other half his new choice is probably 50/50

bucketybuck

ardinn wrote: »

iv no interest in the computer - he picked a, a is still there.

He hasn't picked anything at that point.

He hasn't picked A, it is still just one of two options.

facehugger99

ardinn wrote: »

wasnt guaranteed - but as it transpired, it happened - is mh conditional that it all has to be pre arranged? no! Ill stick with my original assuption

The wrong one?

Sure, you do that.

abff

mikeymouse wrote: »

I agree with your 75% every time he is given the choice to switch,which answers the question.
but he is only given this choice about half the time ,
the other half his new choice is probably 50/50

Monty Hall does not apply in this situation as there is the possibility that A is incorrect and will be eliminated under the 50:50.

mikeymouse

Ok,
supposing it's a Monty Hall with 4 doors,
the only difference is, instead of Monty knowing the contestants choice, it is a computer (which doesn't know his choice) will randomly open two losing doors, regardless of which door the contestant chooses

realdanbreen

mikeymouse wrote: »

Watching 'Who wants to be a millionaire' last night got me thinking;
The guy was on 125k question , he had 2 lifelines; 50/50 and ask the audience.(ignore ask the audience for this)
He had no idea as to the correct answer , but favoured A for no logical reason,
likewise he had no logical reason to discard B C or D.
So he had a 1 in 4 chance of A being right.
He took the 50/50 lifeline and was left with A and B.
So logically that's a 50/50 chance for both A and B,right?
Well,no.
It's not quite Monty Hall , but what are his chances now were he to switch?
(Bear in mind whenever he picks an incorrect answer the computer will discard it
2 out of 3 times)

Was A the correct answer?

mikeymouse

realdanbreen wrote: »

Was A the correct answer?

No B was the answer,
and he also asked the audience who went something like 70/30 in favour of A.

realdanbreen

Can't understand why they start talking about which one they think it might be and then ask the audience. Every member of the audience has to select an answer even if they haven't a clue so the ones with no idea will usually select the answer that the contestant was leaning towards.

mikeymouse

realdanbreen wrote: »

Can't understand why they start talking about which one they think it might be and then ask the audience. Every member of the audience has to select an answer even if they haven't a clue so the ones with no idea will usually select the answer that the contestant was leaning towards.

Yea , there should be a 5th 'I don't know' button .

mikeymouse

abff wrote: »

Monty Hall does not apply in this situation as there is the possibility that A is incorrect and will be eliminated under the 50:50.

If it is incorrect and eliminated then he won't be offered the choice to switch from his original choice.

facehugger99

realdanbreen wrote: »

Can't understand why they start talking about which one they think it might be and then ask the audience. Every member of the audience has to select an answer even if they haven't a clue so the ones with no idea will usually select the answer that the contestant was leaning towards.

This.

If you're going to ask the audience you should do so before you start verbalizing what answer you think it might be.

ardinn

facehugger99 wrote: »

The wrong one?

Sure, you do that.

It isnt - it isnt even monty hall - but the same principles apply - forget the random possibilities and focus on what actually happened and its a 25/75 split!

Peregrinus

mikeymouse wrote: »

If it is incorrect and eliminated then he won't be offered the choice to switch from his original choice.

In Millionaire? Yes, he will. You don't have to make an "original choice" and, if you do, you aren't committed to it until after you have used your 50/50. If this was not so then nobody would ever use 50/50 after making an original choice; it would be just throwing it away.

In Monty Hall the issue cannot arise. Once you have made your initial choice, the elimination is then made only from the other options. The option which is the subject of your initial choice will never be eliminated.

Roger Hassenforder

What was the fecking question?

Ush1

mikeymouse wrote: »

If it is incorrect and eliminated then he won't be offered the choice to switch from his original choice.

Exactly, because there was no original choice. Hence why the odds are 50/50 on the remaining two.

bucketybuck

mikeymouse wrote: »

If it is incorrect and eliminated then he won't be offered the choice to switch from his original choice.

There was no original choice.

He had not yet made any choice, thinking out loud about option A does not mean he had confirmed his choice.

mikeymouse

Peregrinus wrote: »

In Millionaire? Yes, he will. You don't have to make an "original choice" and, if you do, you aren't committed to it until after you have used your 50/50. If this was not so then nobody would ever use 50/50 after making an original choice; it would be just throwing it away.

In Monty Hall the issue cannot arise. Once you have made your initial choice, the elimination is then made only from the other options. The option which is the subject of your initial choice will never be eliminated.

Ok,
supposing it's a Monty Hall with 4 doors,
the only difference is, instead of Monty knowing the contestants choice, it is a computer (which doesn't know his choice) will randomly open two losing doors, regardless of which door the contestant chooses

Peregrinus

ardinn wrote: »

It isnt - it isnt even monty hall - but the same principles apply - forget the random possibilities and focus on what actually happened and its a 25/75 split!

No, it's not. It's impossible to argue that the elimination of options C and D gives you different information about options A and B. It gives you the same information about both of them.

Ush1

mikeymouse wrote: »

Ok,
supposing it's a Monty Hall with 4 doors,
the only difference is, instead of Monty knowing the contestants choice, it is a computer (which doesn't know his choice) will randomly open two losing doors, regardless of which door the contestant chooses

This part is crucial.