Discussion
Or possibly probability puzzle.
Suppose you have a choice of 2 identical boxes both containing cash prizes. All you know is that one contains twice the money of the other.
Obviously it makes no difference to your expected winnings whether you pick A or B and stick with your choice, or pick A and change your mind and go for B, or vice versa, or change your mind twice etc etc.
But suppose that once you've picked one box and found £100 in it you have a once only chance to swap it for the other box which has either £50 or £200 in it.
On the one hand we've already decided it makes no difference to your expected winnings. On the other the swap give you a 50% chance of a £50 loss, and a 50% chance of a £100 gain. Betting £50 on odds of 2 to 1 but with a £150 potential win looks like a good deal. SO why not pick the other box in the first place?
WTF?
Suppose you have a choice of 2 identical boxes both containing cash prizes. All you know is that one contains twice the money of the other.
Obviously it makes no difference to your expected winnings whether you pick A or B and stick with your choice, or pick A and change your mind and go for B, or vice versa, or change your mind twice etc etc.
But suppose that once you've picked one box and found £100 in it you have a once only chance to swap it for the other box which has either £50 or £200 in it.
On the one hand we've already decided it makes no difference to your expected winnings. On the other the swap give you a 50% chance of a £50 loss, and a 50% chance of a £100 gain. Betting £50 on odds of 2 to 1 but with a £150 potential win looks like a good deal. SO why not pick the other box in the first place?
WTF?
It's a bit like the Monty Hall problem, but different 
To answer the 2nd question 1st, why not pick the other box 1st? because you'd have the same dilemma. If you picked the other box first, if it had £200, you have the same gamble, to swap and maybe win £400, or lose and just get £100. If the other box has £50, and you picked it first, you'd be gambling on the other box having £100 or £25. So no matter which one you pick 1st, you've always got that choice.
Regarding the swap, in bookmakers terms it's a good gamble to swap, as you either double or halve your money, so you'll could always win twice as much as you stand to lose, which is good on an evens bet.
So I guess it's down to individual circumstance and how much you could afford to lose and how much you need the money. If I picked a box with £100, I'd happily swap for a possible £200 or £50. But if it was £1m, I'd take the money. Not prepared to risk losing £500K for the chance of a £2m win.
To answer the 2nd question 1st, why not pick the other box 1st? because you'd have the same dilemma. If you picked the other box first, if it had £200, you have the same gamble, to swap and maybe win £400, or lose and just get £100. If the other box has £50, and you picked it first, you'd be gambling on the other box having £100 or £25. So no matter which one you pick 1st, you've always got that choice.
Regarding the swap, in bookmakers terms it's a good gamble to swap, as you either double or halve your money, so you'll could always win twice as much as you stand to lose, which is good on an evens bet.
So I guess it's down to individual circumstance and how much you could afford to lose and how much you need the money. If I picked a box with £100, I'd happily swap for a possible £200 or £50. But if it was £1m, I'd take the money. Not prepared to risk losing £500K for the chance of a £2m win.
Dr Jekyll said:
Or possibly probability puzzle.
Suppose you have a choice of 2 identical boxes both containing cash prizes. All you know is that one contains twice the money of the other.
Obviously it makes no difference to your expected winnings whether you pick A or B and stick with your choice, or pick A and change your mind and go for B, or vice versa, or change your mind twice etc etc.
But suppose that once you've picked one box and found £100 in it you have a once only chance to swap it for the other box which has either £50 or £200 in it.
On the one hand we've already decided it makes no difference to your expected winnings. On the other the swap give you a 50% chance of a £50 loss, and a 50% chance of a £100 gain. Betting £50 on odds of 2 to 1 but with a £150 potential win looks like a good deal. SO why not pick the other box in the first place?
WTF?
How can you mention 3 prizes of £50, £100 and £200 if there are only 2 boxes?Suppose you have a choice of 2 identical boxes both containing cash prizes. All you know is that one contains twice the money of the other.
Obviously it makes no difference to your expected winnings whether you pick A or B and stick with your choice, or pick A and change your mind and go for B, or vice versa, or change your mind twice etc etc.
But suppose that once you've picked one box and found £100 in it you have a once only chance to swap it for the other box which has either £50 or £200 in it.
On the one hand we've already decided it makes no difference to your expected winnings. On the other the swap give you a 50% chance of a £50 loss, and a 50% chance of a £100 gain. Betting £50 on odds of 2 to 1 but with a £150 potential win looks like a good deal. SO why not pick the other box in the first place?
WTF?
mrmr96 said:
Dr Jekyll said:
Or possibly probability puzzle.
Suppose you have a choice of 2 identical boxes both containing cash prizes. All you know is that one contains twice the money of the other.
Obviously it makes no difference to your expected winnings whether you pick A or B and stick with your choice, or pick A and change your mind and go for B, or vice versa, or change your mind twice etc etc.
But suppose that once you've picked one box and found £100 in it you have a once only chance to swap it for the other box which has either £50 or £200 in it.
On the one hand we've already decided it makes no difference to your expected winnings. On the other the swap give you a 50% chance of a £50 loss, and a 50% chance of a £100 gain. Betting £50 on odds of 2 to 1 but with a £150 potential win looks like a good deal. SO why not pick the other box in the first place?
WTF?
How can you mention 3 prizes of £50, £100 and £200 if there are only 2 boxes?Suppose you have a choice of 2 identical boxes both containing cash prizes. All you know is that one contains twice the money of the other.
Obviously it makes no difference to your expected winnings whether you pick A or B and stick with your choice, or pick A and change your mind and go for B, or vice versa, or change your mind twice etc etc.
But suppose that once you've picked one box and found £100 in it you have a once only chance to swap it for the other box which has either £50 or £200 in it.
On the one hand we've already decided it makes no difference to your expected winnings. On the other the swap give you a 50% chance of a £50 loss, and a 50% chance of a £100 gain. Betting £50 on odds of 2 to 1 but with a £150 potential win looks like a good deal. SO why not pick the other box in the first place?
WTF?
On a purely mathematical basis you should always swap. The question only has any real meaning if the sums involved are large enough that you wouldn't want to lose half of what you have, sprinkled with a bit of your own attitude to gambling.
So for me, well £50 to win £25 or £100 is a no-brainer. Make it £1000 and I'll take it all day long. However, a friend of mine of a similar financial standing would take the punt at £1000. He'd probably take a punt at £10k too.
So for me, well £50 to win £25 or £100 is a no-brainer. Make it £1000 and I'll take it all day long. However, a friend of mine of a similar financial standing would take the punt at £1000. He'd probably take a punt at £10k too.
budfox said:
On a purely mathematical basis you should always swap. The question only has any real meaning if the sums involved are large enough that you wouldn't want to lose half of what you have, sprinkled with a bit of your own attitude to gambling.
So for me, well £50 to win £25 or £100 is a no-brainer. Make it £1000 and I'll take it all day long. However, a friend of mine of a similar financial standing would take the punt at £1000. He'd probably take a punt at £10k too.
Why ahold you always swap on a 'mathematical basis'?So for me, well £50 to win £25 or £100 is a no-brainer. Make it £1000 and I'll take it all day long. However, a friend of mine of a similar financial standing would take the punt at £1000. He'd probably take a punt at £10k too.
If you stick to the original box then you have 100% chance of winning £100
If you switch, you have 50% chance of getting £50 and 50% chance of getting £200. On average, switching will deliver £125.
It makes sense to switch and over time, given multiple opportunities, it's a no brainer to switch.
If you switch, you have 50% chance of getting £50 and 50% chance of getting £200. On average, switching will deliver £125.
It makes sense to switch and over time, given multiple opportunities, it's a no brainer to switch.
This is subjective probability where you have to assign a utility function to the certainty and uncertainty.
Subjective probability is NOT classical probability.
The key is with the utility functions, as the "usefullness" of the certain outcome, versus the "usefullness" of the uncertain outcome, is all down to subjective risk.
If in doubt perform the same exercise with values 100x larger, peoples risk profiles will alter.
This is NOT stochastic mathematics. This is psychological studies of behavioural risk.
Fortunately I have been reading up on this topic!
Subjective probability is NOT classical probability.
The key is with the utility functions, as the "usefullness" of the certain outcome, versus the "usefullness" of the uncertain outcome, is all down to subjective risk.
If in doubt perform the same exercise with values 100x larger, peoples risk profiles will alter.
This is NOT stochastic mathematics. This is psychological studies of behavioural risk.
Fortunately I have been reading up on this topic!
mrmr96 said:
stefd said:
If you switch, you have 50% chance of getting £50 and 50% chance of getting £200. On average, switching will deliver £125.
.
You sure about that?.
TwigtheWonderkid said:
Seems spot on to me. If you switch on £100, half the time you'll end up with £50, and half the time you'll end up with £200. So the average prize for switching is £125. If you don't switch, you will always win £100. Therefore switching, on average, delivers a better yield.
But what blows my mind is that once you've switched the same argument applies for switching back. It makes no difference which one you pick, but switching seems good.TwigtheWonderkid said:
mrmr96 said:
stefd said:
If you switch, you have 50% chance of getting £50 and 50% chance of getting £200. On average, switching will deliver £125.
.
You sure about that?.
This is NOT MATHS.
The experiment is not repeated, therefore what happens on average is not important. What happens in your specific case is important. Therefore your minimax (worst case scenario) is either £50 or £100. Therefore what will influence the choice is the value assigned to £100/£50 and the acceptable worst case scenario.
Minimax can be considered to be a subset of game or decision theorys.
TwigtheWonderkid said:
mrmr96 said:
stefd said:
If you switch, you have 50% chance of getting £50 and 50% chance of getting £200. On average, switching will deliver £125.
.
You sure about that?.
(I'm not bring an arse, I'm trying to guide you guys to the answer!
)Here is my answer:
The point is that you're not told "the other box has either £50 or £200." You're told "one box has double the value of the other". At the point you made your original choice it was 50/50 whether you'd choose the more valuable box. Opening your chosen box has no influence over this. It's still 50/50 that you already have the more valuable box so it should be clear that there is no benefit in switching. (Indeed I think you already know this is correct, its the reason why you know that your 'expected value' calc cannot be correct.)
The place your logic falls down is in trying to apply an expected value calculation to a scenario where it cannot be used.
Expected values can be calculated by considering the sum of all values multiplied by their probabilities.
Having opened a box with £100 in it is not a 50/50 chance that the other box contains £50 or £200. Rather from this point there can only be one outcome to this game. It's either a 100% chance of £50, or a 100% chance of £200. That is certain because there is a box there with only one value inside it. That value cant and won't change DURING THIS GAME. (For it to have another value you must start another game.) This is why you can't apply expected value calculations, as you are partway through a PARTICULAR game.
The correct way to apply expected values would be to consider the two possible outcomes to each strategy:
Switching:
1 low box and switch
2 high box and switch
Expected value = 2n/2 + n/2
Sticking:
1 low box and stick
2 high box and stick
Expected value = n/2 + 2n/2
There is no benefit to switching.
Hope that helps.
The point is that you're not told "the other box has either £50 or £200." You're told "one box has double the value of the other". At the point you made your original choice it was 50/50 whether you'd choose the more valuable box. Opening your chosen box has no influence over this. It's still 50/50 that you already have the more valuable box so it should be clear that there is no benefit in switching. (Indeed I think you already know this is correct, its the reason why you know that your 'expected value' calc cannot be correct.)
The place your logic falls down is in trying to apply an expected value calculation to a scenario where it cannot be used.
Expected values can be calculated by considering the sum of all values multiplied by their probabilities.
Having opened a box with £100 in it is not a 50/50 chance that the other box contains £50 or £200. Rather from this point there can only be one outcome to this game. It's either a 100% chance of £50, or a 100% chance of £200. That is certain because there is a box there with only one value inside it. That value cant and won't change DURING THIS GAME. (For it to have another value you must start another game.) This is why you can't apply expected value calculations, as you are partway through a PARTICULAR game.
The correct way to apply expected values would be to consider the two possible outcomes to each strategy:
Switching:
1 low box and switch
2 high box and switch
Expected value = 2n/2 + n/2
Sticking:
1 low box and stick
2 high box and stick
Expected value = n/2 + 2n/2
There is no benefit to switching.
Hope that helps.
HowMuchLonger said:
TwigtheWonderkid said:
mrmr96 said:
stefd said:
If you switch, you have 50% chance of getting £50 and 50% chance of getting £200. On average, switching will deliver £125.
.
You sure about that?.
This is NOT MATHS.
The experiment is not repeated, therefore what happens on average is not important.
Bottom line is this. You open box A, it has £100. Box B therefore has £200 or £50. Some days it'll be £200, some days it'll be £50. Assuming the amount in box B is either allocated to be double or half on a random chance basis, then by swapping, the average winnings of 100 players over 100 occasions will be £125. The average winnings achieved by the non swappers will be £100.
Of course you could open box B first. It may have £200. But then box A either has £400 or £100, a gain of £200 or a loss of £100, so you have the same dilemma. If box B has £50, box A has either £100 or £25. A gain of £50 or a loss of £25. You always win more when you get it right than you lose when you get it wrong
Edited by TwigtheWonderkid on Wednesday 5th February 11:26
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