A question of Probability

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A little workout for your brain......

Imagine I offered you three identical boxes and said one contains a £million, choose one.

You make your choice pointing it out to me, say A, B or C.

I, knowing which one contains the cash, then removed an empty one.

Then I offered you the opportunity to stick or switch, which would you do?

Please don’t look up the answer on the net and I’ll try to private message you with the correct answer so as not to spoil it for others. If it sounds incredibly easy it isn’t, it has perplexed many bright minds since first being posed in the 19th Century.

So stick or switch?
 

ukzero1

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At this time in a morning??...good grief. Hmmm..stick with the baon sarny? or switch to the bacon and egg? That's as far as I've got just yet. I need more coffee.
 
D

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At this time in a morning??...good grief. Hmmm..stick with the baon sarny? or switch to the bacon and egg? That's as far as I've got just yet. I need more coffee.
Apologies you’re allowed to hold back til the brown sauce kicks in!
 

mickthechippy

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A little workout for your brain......

Imagine I offered you three identical boxes and said one contains a £million, choose one.

You make your choice pointing it out to me, say A, B or C.

I, knowing which one contains the cash, then removed an empty one.

Then I offered you the opportunity to stick or switch, which would you do?

Please don’t look up the answer on the net and I’ll try to private message you with the correct answer so as not to spoil it for others. If it sounds incredibly easy it isn’t, it has perplexed many bright minds since first being posed in the 19th Century.

So stick or switch?


stick with my first choice

I have learnt in my long life, if you have a gut feeling, never ignore it
 

Scribe

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Stick - Fortune favours the brave. That is as long as I didn't pick the empty box that had already been removed in the first place ! :)

I have a feeling this is similar to the flip a coin probability question which on the face of it is 50/50 heads or tails,however it isn't one will come out higher than the other.
 

Scribe

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@Deleted Member I am reciting from memory but as I recall on tests carried out tails predominantly fell face up more often. this could do with the weight of the coin being un-equal between the two - as I said it was something I read in a scientific paper a very long time ago.
 
D

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@Deleted Member I am reciting from memory but as I recall on tests carried out tails predominantly fell face up more often. this could do with the weight of the coin being un-equal between the two - as I said it was something I read in a scientific paper a very long time ago.
Obviously if the pattern on the coin has a weighting caused by the design or tolerances in manufacturing, both are of which are likely in the real world, then that would sway the result over statistically significant testing.

However, in probability, we assume the coin is perfect and therefore the probability is also perfect at 50:50.
 

lp1886

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If it’s down to probability then by switching puts the odds in my favour. But that’s not to say it’s a guarantee of winning the money. I think I remember this being explained in a film.
 

spanky

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Not this old chestnut.... switch. What with conditional probabilities and all that.

I even remember modelling this out a good few years back using a random numbers and hundreds of simulations to prove the point
 
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spanky

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For those of you who want to think it through a bit further:

1565166936564.png

So, on average switching increases the odds of winning from 33% to 67%
 

Markywhizz

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Spanky is correct but the other argument which is equally strong is that with only two boxes left the probability is 50% so it doesn’t matter which you choose. It is therefore a paradox. Since in the second argument it makes no difference you would be best to switch anyway.
 

spanky

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Spanky is correct but the other argument which is equally strong is that with only two boxes left the probability is 50% so it doesn’t matter which you choose. It is therefore a paradox. Since in the second argument it makes no difference you would be best to switch anyway.

I'm afraid to say your statement 'The other argument which is equally strong' is incorrect. If you had already chosen the box with the money you have a 50:50 chance of being correct, but if you have chosen an empty box you are guaranteed to win if you switch (because a losing box has been removed). This is the concept behind conditional probability.

To give you a more extreme example... suppose there were 100 boxes and only one had money and you have to chose one. Then 98 losing boxes were removed would you stick or switch? You have a 1 in 100 chance if you stick, but a 50:50 if you switch? What would you do, stick or switch?
 
D

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I'm afraid to say your statement 'The other argument which is equally strong' is incorrect. If you had already chosen the box with the money you have a 50:50 chance of being correct, but if you have chosen an empty box you are guaranteed to win if you switch (because a losing box has been removed). This is the concept behind conditional probability.

To give you a more extreme example... suppose there were 100 boxes and only one had money and you have to chose one. Then 98 losing boxes were removed would you stick or switch? You have a 1 in 100 chance if you stick, but a 50:50 if you switch? What would you do, stick or switch?
Correct and expertly explained!
 

Markywhizz

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I'm afraid to say your statement 'The other argument which is equally strong' is incorrect. If you had already chosen the box with the money you have a 50:50 chance of being correct, but if you have chosen an empty box you are guaranteed to win if you switch (because a losing box has been removed). This is the concept behind conditional probability.

To give you a more extreme example... suppose there were 100 boxes and only one had money and you have to chose one. Then 98 losing boxes were removed would you stick or switch? You have a 1 in 100 chance if you stick, but a 50:50 if you switch? What would you do, stick or switch?

If 98 empty boxes have been removed and you know they were empty the 1 in 100 chance has gone and the odds are recalculated you are back to a 50% chance whether you switch or stick surely.
 
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