Can you solve the false positive riddle? – Alex Gendler


Mining unobtainium is hard work. The rare mineral appears
in only 1% of rocks in the mine. But your friend Tricky Joe
has something up his sleeve. The unobtainium detector he’s been
perfecting for months is finally ready. The device never fails
to detect unobtainium if any is present. Otherwise, it’s still highly reliable, returning accurate
readings 90% of the time. On his first day trying
it out in the field, the device goes off, and
Joe happily places the rock in his cart. As the two of you head back to camp
where the ore can be examined, Joe makes you an offer: he’ll sell you the ore for just $200. You know that a piece of unobtanium
that size would easily be worth $1000, but any other minerals
would be effectively worthless. Should you make the trade? Pause here if you want
to figure it out for yourself. Answer in: 3 Answer in: 2 Answer in: 1 Intuitively, it seems like a good deal. Since the detector
is correct most of the time, shouldn’t you be able
to trust its reading? Unfortunately, no. Here’s why. Imagine the mine
has exactly 1,000 pieces of ore. An unobtainium rarity of 1% means that there are only 10 rocks
with the precious mineral inside. All 10 would set off the detector. But what about the other 990
rocks without unobtainium? Well, 90% of them,
891 rocks, to be exact, won’t set off anything. But 10%, or 99 rocks,
will set off the detector despite not having unobtanium, a result known as a false positive. Why does that matter? Because it means that all in all, 109 rocks will have
triggered the detector. And Joe’s rock could be any one of them, from the 10 that contain the mineral to the 99 that don’t, which means the chances of it containing
unobtainium are 10 out of 109 – about 9%. And paying $200 for a 9%
chance of getting $1000 isn’t great odds. So why is this result so unexpected, and why did Joe’s rock seem
like such a sure bet? The key is something called
the base rate fallacy. While we’re focused on the relatively
high accuracy of the detector, our intuition makes us forget to account for how rare the unobtanium
was in the first place. But because the device’s error rate of 10% is still higher than
the mineral’s overall occurrence, any time it goes off is still more likely
to be a false positive than a real finding. This problem is an example
of conditional probability. The answer lies neither in the overall
chance of finding unobtainium, nor the overall chance
of receiving a false positive reading. This kind of background information
that we’re given before anything happens is known as unconditional,
or prior probability. What we’re looking for, though,
is the chance of finding unobtainium once we know that the device did
return a positive reading. This is known as the conditional,
or posterior probability, determined once the possibilities have
been narrowed down through observation. Many people are confused
by the false positive paradox because we have a bias
for focusing on specific information over the more general, especially when immediate decisions
come into play. And while in many cases
it’s better to be safe than sorry, false positives can have
real negative consequences. False positives in medical testing
are preferable to false negatives, but they can still lead to stress or
unnecessary treatment. And false positives in mass surveillance can cause innocent people to be
wrongfully arrested, jailed, or worse. As for this case, the one thing
you can be positive about is that Tricky Joe is trying
to take you for a ride.

100 thoughts on “Can you solve the false positive riddle? – Alex Gendler

  1. Sign up for free at https://brilliant.org/TedEd/, and Brilliant will email you the solution to the bonus riddle! Also, the first 833 of you who sign up for a PREMIUM subscription will get 20% off the annual fee. Riddle on, friends!

  2. I would just buy his machine and copy the materials to make my own machine and shoot the rocks myself x10 times so I can verify. Solved.

  3. Answer for the bonus riddle:
    a pack cards has 13 different cards of 4 shapes each = 52 cards. So u can make 6 pairs with each number card. For ex: with 4 queens u can make 6 pairs (spade&heart, spade&clubs, spade&diamond, heart&clubs, heart&diamond, clubs&diamond). Thus in total there is a probability of 1/78 (6pairs * 13 cards= 78). ur opponent already has 2 queens which means there are only 2 queens left in the remaining deck eliminating 5 pairs of queens above said. that will increase the probability to 1/73 (78-5). In the next scenario ur opponent has 1Queen, 1five. So the number of pairs u can make with remaining cards is 72 pairs (78-3-3). So the probability of getting a pair is better in second scenario which is 1/72 (As we all know that 1/72 is better than 1/73). Like if anyone understand this

  4. People have a habit of looking over specific things, oh you mean the fact that he’s tricky joe or we couldn’t of scanned this twenty time(or maybe it mistakes cause… it gives off a certain pulse or something), or how I’m a poor miner and I don’t have 200% or…. ummmm…. math

  5. LOL as I work in the Department of Epidemiology, I immediately thought of this as the positive predictive value and gave the instant result :))

  6. OH MY GOD I ACTUALLY SOLVED SOMETHING!! The first scenario would be the better chance I’m assuming because that only takes away one possible pair out of the deck, the Queens. But with the second scenario, it takes away two possible pairs, queen and a 5 since she has one of each.

  7. Or you could shoot the device 2 or 3 more times at the rock and there is a SUPER small chance then that you're getting a bad deal, because it ALWAYS detects actual Unobtanium.

  8. wouldn't you still be checking 90% less rocks? so instead of unobtanium being 1 percent of the rocks you carry its about 9%, isn't it a reusable tool? it would still greatly increase the chances of you finding unobtanium.

  9. If Ik the ore is in 10% or rocks and the size of the rock if it was pure would be $1,000 at best odds that’s still only $100 of ore plus the cost of labor on me! That alone is enough to sway me away, plus given the ore could have been picked up from another sample I would have him retest it a few times outside to make the probability that it is in there higher. if positive best I would give him is $50

  10. Soooo, ya sayin’ diamonds, obsidian, emerald, sapphire, aquamarine, opal and all the other precious rocks are worthless???

  11. This is honestly easier than the passcode riddle. Its so intuitive. I didnt even need to think to solve this, and i got it right.

  12. It's weird, because my immediate mathematics tells me "it's only worth it if it's a $20 price tag", and I can't remember what those mathematics are after seeing the video.

  13. This make no sense you said the laser gun could find unobtanium without failure but then you said it would have a nine percent chance of it finding it

  14. Only put it in cart if it returned a positive reading four consecutive times — that increases the accuracy from 90% to 99.99%

  15. Tricky joe- you know who i am?
    Me-no,who are you?
    Tricky joe- i am joe
    Me-wait,who is joe?
    Tricky joe- JOE ROCK

    brought to you by: joe mama jokes co.

  16. HOWEVER…. joe was offering 200$ when its supposed to be worth 1000$
    as logic would indicate joe is possibly scamming you because he knows that the rock is worthless.

  17. Would joes rock technically not be worth 800 cuz youre paying 200 dollars for something possibly worth 1000

  18. This is the exact reason that urine tests are so unreliable. When they say that it is accurate 97% of the time, they do not take into account the false positives and the false negatives in their tests.

  19. the answer to the second riddle is that it's more likely if she has 2 queens. because there's still a pair of queens in the deck, meanwhile with the q and 5, she has cards which take from 2 different numbers, which means there's less chance for a pair from both the queen and the 5

  20. Threaten to break his skull and mine the rock for free for the unobtainium
    If you don’t get it, break his skull
    If you do, give him a concussion and steal the ore

  21. just do it multiple times, it will have a real small chance of false reading it 10 times in a row, it will make it real easy to detect unobtanium and detect false readings

  22. Parameters:
    1% contain x
    99% contain y
    90% chance of finding x
    10% chance of false positive

    Out of 100 rocks, 11 say x (x is found 100% when it actually is x, and 10% are false positives)
    Therefore, there is a 9% chance out of 11 anomalies that x is true.

    Expected value for x: 0.09 * 1000
    = $90

    $200 for purchase, so on average net loss $110 occurs.

    Therefore, it is definitely NOT worth taking Tricky Joe's offer.

    Duration of completion: 78 seconds.

    That was a fun riddle, but it was mostly statistics, no lateral thinking required.

  23. If you shoot it again does it give the same answer or reroll the odds. If its the latter, I will just shoot it a few more times.

  24. Bonus riddle. Through all 4 suits there are 26 pairs all together. If she took two queens that would equal 25 more pairs. If she took 1 queen and 1 five. Then she took 0.5 pairs and 0.5 pairs that add up to 1… so the answer is that they have the same odds

  25. Just shoot it twice then, if it goes off twice then it has a very high chance of being unobtainium, if it goes off once but in the other shot it doesnt go off then its not it chief

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