Poll of the Day > A probability question

(+) Yeah! (+)

Your situation:

You are a miner, searching for rare minerals. Typically this is long and exhausting work - you carve out a large chunk of rock and haul it to the surface for analysis and processing, which can only be done using large, specialized equipment. Worse, most of the rocks are worthless - just 1% of all rocks have valuable minerals in them. However, a friend of yours has been working on a better way. One day, he offers to show you a new device he has invented which he is convinced will make mining a breeze.

The device is a small, portable device that can scan a mined piece of rock in seconds and deliver a preliminary analysis on whether or not the rock contains rare minerals.
-If the rock contains rare minerals, the device will always return a positive reading.
-If the rock does not contain rare minerals, the device is less precise. 90% of the time it will correctly identify that the rock has no rare minerals; however, 10% of the time it will return a false positive reading and report that the rock contains rare minerals.

You and your friend go down into the mines. As you are working, he is busy scanning. Suddenly, his detector goes off - the rock he is scanning has a positive reading. Since he discovered it, the rock is, by all rights, your friend's claim. However, he makes you an offer - he will sell you the rock for $500. The market value of the rock, if it contains rare minerals, would be about $1000; if it contains no minerals, it is effectively worthless.

Should you take your friend's offer?
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1% vs 90%

???

No, the risk-to-reward ratio is too low. While the error rate is reasonably low, I'd lose everything if it fails in a deal that would only double my money. If I'm putting down $500, I'd want at least the chance of a tenfold return.
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darkknight109 posted...

You are a miner, searching for rare minerals.

darkknight109 posted...

most of the rocks are worthless - just 1% of all rocks have valuable minerals in them

Sounds like it's time to polish up the ol' resume.
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Zeus posted...

No, the risk-to-reward ratio is too low. While the error rate is reasonably low, I'd lose everything if it fails in a deal that would only double my money. If I'm putting down $500, I'd want at least the chance of a tenfold return.

10% vs 90%? And the alternative is 1% vs 99%?

Well, someone mathematically as adept as @Sahuagin correct me if I'm wrong, but I'm just going to say no, based on the expected value of the venture.

Denote:

M = rock has minerals
M` = rock does not have minerals
+ = test positive

Given:
P(+|M) = 1
P(+|M`) = 0.1
P(M) = 0.01

So the expected value of the venture would be:

E(M) = P(+|M)P(M)*(1000) + P(+|M`)P(M`)*0 = (1)(0.01)(1000) = 10 < 500

You'd be losing an average of $490 to your "friend", and so you wouldn't take the deal. It's been a long time since I've done probability, so forgive me for my poor math.
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darkknight109 posted...

-If the rock contains rare minerals, the device will always return a positive reading.
-If the rock does not contain rare minerals, the device is less precise. 90% of the time it will correctly identify that the rock has no rare minerals; however, 10% of the time it will return a false positive reading and report that the rock contains rare minerals.

No: I continue hating my life and I effectively earn an incredibly inconsistent $10 per rock ($1000 * 1%)
Yes: I earn consistent $480 a rock, I increase my profits by %4800 with literally no downside. ($500 * 90%)

You left out some details. Will my friend continue to do this? If this is a one-off deal, no. In this life, money is something hard to come by and I don't want the 10% chance of starving to death in this apparently third world country. If he'll do it for me repeatedly, it's a no brainer.

Actually, he's got a good system. He can have multiple customers, and make up for his own 48% loss my more than doubling the amount of rocks he could sell without other people carrying rocks out for him.

EclairReturns posted...

You'd be losing an average of $490 to your "friend", and so you wouldn't take the deal. It's been a long time since I've done probability, so forgive me for my poor math.

Actually, I made that mistake, but the false negative is 100% a false negative, not a 50%. So that number is actually $480.

Actually, the number would fall back to the default 1%, so $480 + $0.002.

Those would be your profits, your friend would be gaining $1000 - $480 ($520) per trade (not taking into account the money he could make taking the rocks out himself)

I don't have many friends.

Yellow posted...

You left out some details. Will my friend continue to do this? If this is a one-off deal, no. In this life, money is something hard to come by and I don't want the 10% chance of starving to death in this apparently third world country. If he'll do it for me repeatedly, it's a no brainer.

Statistically speaking, whether your friend only offers you the deal once or repeatedly does not change whether or not it's a good deal.
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darkknight109 posted...

Statistically speaking, whether your friend only offers you the deal once or repeatedly does not change whether or not it's a good deal.

Well, if it's not repetitive, it's not a statistic, is it?

Yellow posted...

darkknight109 posted...

Statistically speaking, whether your friend only offers you the deal once or repeatedly does not change whether or not it's a good deal.

Well, if it's not repetitive, it's not a statistic, is it?

Sure it is. Statistics don't need to be repeated to be statistics.

A coin toss has a 50% chance of landing on heads, regardless of whether you flip it one times or one hundred.
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I just bludgeon my friend over the head, I get the device and the rock, ok?
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darkknight109 posted...

Yellow posted...

You left out some details. Will my friend continue to do this? If this is a one-off deal, no. In this life, money is something hard to come by and I don't want the 10% chance of starving to death in this apparently third world country. If he'll do it for me repeatedly, it's a no brainer.

Statistically speaking, whether your friend only offers you the deal once or repeatedly does not change whether or not it's a good deal.

It changes the viability.
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Statistics even out over time. That is incredibly relevant in whether or not I'll take your deal.

I would rather have a 100% chance of having a pebble thrown at me than a 1% chance of having a boulder thrown at me.

Kyuubi4269 posted...

It changes the viability.

Viability, again, has nothing to do with whether or not it's statistically a good deal.

As an example, if I offered you a lotto ticket that had a 40% chance of winning a million dollars, but it cost you $200k, statistically speaking that's a good deal. You, personally, may not want to take that offer due to the high buy-in cost and considerable chance of losing that investment, but that doesn't change that the statistics still indicate it's a favourable deal.

Yellow posted...

Statistics even out over time. That is incredibly relevant in whether or not I'll take your deal.

I didn't ask whether you would take the deal, I asked whether it was a good deal.

There's a difference.
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darkknight109 posted...

Viability, again, has nothing to do with whether or not it's statistically a good deal.

As an example, if I offered you a lotto ticket that had a 40% chance of winning a million dollars, but it cost you $200k, statistically speaking that's a good deal. You, personally, may not want to take that offer due to the high buy-in cost and considerable chance of losing that investment, but that doesn't change that the statistics still indicate it's a favourable deal.

If it's not viable, it's statistically worse as you will lose more than the failure result if you fail.
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If it's a one-off deal, then it's bad. Like, 99% bad.
If it's a partnership, then it's very, very good. 100% good.

Is that what you wanted?

If it's a 50% chance of either type of deal, because you won't tell me, then I upgrade that deal to 98% bad. I don't accept a 5% chance of starving for money.

You can't staple a single value to a deal regardless of the situation. That kind of thinking hurts my brain.

Regardless, if you want the imaginary quantum figure;

Yellow posted...

No: I continue hating my life and I effectively earn an incredibly inconsistent $10 per rock ($1000 * 1%)

Yellow posted...

I earn consistent $480 a rock, I increase my profits by %4800 with literally no downside. ($500 * 90%)

$10 : $480
1 : 48
96% Good
4% Bad

Ow.

I hope you see why this doesn't make sense.

Kyuubi4269 posted...

If it's not viable, it's statistically worse as you will lose more than the failure result if you fail.

In that instance it is worse, not statistically worse.

You are altering the problem by adding in additional variables when you do that.

Yellow posted...

If it's a one-off deal, then it's bad. Like, 99% bad.
If it's a partnership, then it's very, very good. 100% good.

Is that what you wanted?

You can interpret it that way if you like.

As a bit of a spoiler, at least part of your answer is wrong.
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If we tested 1000 rocks, we would expect 10 of them to actually contain rare minerals on a positive result, while 99 of them would not on a positive result. Assuming we bought all of the rocks that had a positive result, we would spend $54,500, and only receive $10,000 when we sell the 10 rocks that actually contain rare minerals.

That's a net expected loss of $44,500. Our average expected loss over all the 109 rocks we would buy out of the 1000 that were tested is ~$408.26 per rock.

This is a terrible deal.
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darkknight109 posted...

-If the rock does not contain rare minerals, the device is less precise. 90% of the time it will correctly identify that the rock has no rare minerals; however, 10% of the time it will return a false positive reading and report that the rock contains rare minerals.

Fuck, I didn't realize that margin of error scaled with the amount of rocks you would have to go through. It's a bad deal in all situations.

https://imgur.com/H39CG7G

I am ashamed.

dainkinkaide posted...

If we tested 1000 rocks, we would expect 10 of them to actually contain rare minerals on a positive result, while 99 of them would not on a positive result. Assuming we bought all of the rocks that had a positive result, we would spend $54,500, and only receive $10,000 when we sell the 10 rocks that actually contain rare minerals.

That's a net expected loss of $44,500. Our average expected loss over all the 109 rocks we would buy out of the 1000 that were tested is ~$408.26 per rock.

This is a terrible deal.

Youre assuming all rocks are the same size and offer the same profit as the specific rock in question.

Somebody offers you a 90% chance to double your 500$. If youre wrong, you lose your 500$.

Do you take it?

It depends on the situation, it depends on whats going on in your life, it depends on how much 500$ means to you in that moment

MICHALECOLE posted...

Youre assuming all rocks are the same size and offer the same profit as the specific rock in question.

That's totally irrelevant; the probability of the specific rock chosen remains the same.

The rock has a probability of 10/109 of actually containing rare minerals, and a probability of 99/109 of not containing rare minerals.

Ergo, the expected result for this one rock is 10/109*($1000-$500) + 99/109*($0-$500), which is still a net expected loss of ~$408.26.
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Yeah, and for every rock you go through the odds of you getting a false dud are so high that it's basically a lottery ticket.

EclairReturns posted...

someone mathematically as adept as @Sahuagin

I dunno why you think that, in some ways I hate math, though I do like solving problems

in real life, if someone offers you a deal to make money, but only if you first give them money, you should usually if not always say no

otherwise I would have started off by thinking it sounded like a good deal

dainkinkaide posted...

If we tested 1000 rocks, we would expect 10 of them to actually contain rare minerals on a positive result, while 99 of them would not on a positive result. Assuming we bought all of the rocks that had a positive result, we would spend $54,500, and only receive $10,000 when we sell the 10 rocks that actually contain rare minerals.

-If the rock does not contain rare minerals, the device is less precise. 90% of the time it will correctly identify that the rock has no rare minerals; however, 10% of the time it will return a false positive reading and report that the rock contains rare minerals.

right, so for 10% of the bad rocks (99% of the rocks), it will say it's good when it isn't. so 990 bad rocks * 10% = 99 false positives out of 1000.

so out of 1000 rocks:
99 false positives
10 true positives
891 true negatives

the question was worded to make it sound like it had only a 10% chance of being wrong when it identified a good rock, but no it's more like a 9.17% chance of being right.

the gizmo would still be very beneficial, because when it tells you a rock is bad, it really is bad, so it saves you from manually checking almost 90% of the rocks. but the proportion of good rocks to bad in the ones it thinks are good is too low to blindly accept them as being good. they require manual inspection first.
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A more extreme example of this is a device that gives you a 10% false-positive on a winning lottery ticket.

Statistically not worth it.

I'd pass.
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Im using ballpark figures here, but if 99% of the rocks are common, and the scanner misreads 10% of the common rocks as rare, then arent we looking at roughly 90% of the positive reads to be incorrect?

Because say for simplicitys sake, every 100th rock you find has rare materials. That would mean the first 99 rocks are common. And the scanner would make the following reads:

Rocks 1-10 = 1 false positive read
Rocks 11-20 = 1 false positive read
Rocks 21-30 = 1 false positive read
Rocks 31-40 = 1 false positive read
Rocks 41-50 = 1 false positive read
Rocks 51-60 = 1 false positive read
Rocks 61-70 = 1 false positive read
Rocks 71-80 = 1 false positive read
Rocks 81-90 = 1 false positive read
Rocks 91-100 = 1 true positive read
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MICHALECOLE posted...

Somebody offers you a 90% chance to double your 500$. If youre wrong, you lose your 500$.

Do you take it?

It depends on the situation, it depends on whats going on in your life, it depends on how much 500$ means to you in that moment

I'd take those odds/chances/risks/I don't know the mathematically correct term for this

Sahuagin posted...

I dunno why you think that

I could have sworn that when I first posted on this board, that someone proved to me that the product of two squares is the square of the products. Maybe it was a different bloke...
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MICHALECOLE posted...

Zeus posted...

No, the risk-to-reward ratio is too low. While the error rate is reasonably low, I'd lose everything if it fails in a deal that would only double my money. If I'm putting down $500, I'd want at least the chance of a tenfold return.

10% vs 90%? And the alternative is 1% vs 99%?

No, I'd want the same odds. Keep in mind I'm the one putting up actual money and, if he's wrong, he still leaves with money. And, honestly, if I was the guy who found the rock, I'd probably take the $500 for it.

The Popo posted...

Im using ballpark figures here, but if 99% of the rocks are common, and the scanner misreads 10% of the common rocks as rare, then arent we looking at roughly 90% of the positive reads to be incorrect?

Because say for simplicitys sake, every 100th rock you find has rare materials. That would mean the first 99 rocks are common. And the scanner would make the following reads:

Rocks 1-10 = 1 false positive read
Rocks 11-20 = 1 false positive read
Rocks 21-30 = 1 false positive read
Rocks 31-40 = 1 false positive read
Rocks 41-50 = 1 false positive read
Rocks 51-60 = 1 false positive read
Rocks 61-70 = 1 false positive read
Rocks 71-80 = 1 false positive read
Rocks 81-90 = 1 false positive read
Rocks 91-100 = 1 true positive read

This is why I wish I had taken that statistics course.
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darkknight109 posted...

Since he discovered it, the rock is, by all rights, your friend's claim.

I don't think that's how that works.
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