The underlying maths has many other applications of course.
For example, suppose you're told that a facial recognition system has a 99% accuracy rate when comparing 2 head shots to decide whether those 2 head shots are the same person or not. We'll assume for simplicity that we have an equal rate of false positives (falsely saying it is the same person) and false negatives (falsely saying it is not the same person).
This sounds pretty good, and if you were to use it to ensure only the employees of a small company could enter its premises it would probably work well enough, at least if some allowance is made for the occasional mis-identification.
Now, we know that there's a 99/100 chance of correctly recording if 2 head shots are of the same person. But imagine using this to scan headshots taken (via CCTV) of members of the public as they walk down the streets of a busy city to check them against headshots of wanted criminals. Now the odds of getting things wrong become pertinent to your freedom to walk the streets unaccosted by the police.
How many comparisons would you need to do before the probability drops below 50%? The answer is 69. If you compare hundreds or thousands of people against the mugshots of wanted criminals you'll end up accusing lots of random civilians of being criminals and may well miss some of the wanted criminals along the way.
But suppose we raise the accuracy to 99.9%? The point at which the probability drops below 50% is 693. At 99.99% accuracy, the figure for dropping below 50% accuracy is 6,934. This application of facial recognition is going to need a very high accuracy to avoid lots of innocent people being flagged incorrectly as wanted criminals.
Now maybe there are ways around this by e.g. treating the initial match as a trigger for a deeper check of the person's identity to finally settle whether they're really the wanted criminal or not, but I'm sceptical of this kind of use of facial recognition because the maths indicate that an impressive level of pairwise-accuracy can nevertheless yield lots of error in such settings.
My very feeble old-man brain remembered the group size as 13, obviously wrong. But in any case, I always wondered how a small group could give me winning odds. I’ll probably stick with groups of 25 to get me closer to something like Vegas roulette house odds. Thanks for the explainer.
In my case, the leap from "does anyone in this crowd share a birthday? Want to bet?" to "does anyone in this crowd share MY birthday..." happens too fast for it to be a matter of simply substituting an easier question for a harder one. At first, I wanted to blame my listening skills, and then I wanted to blame my early training in memorizing the multiplcation tables ('what's 6 x 7? Quick! Wow, you need to be able to answer faster than that.'), but now I'm just not sure why it is so easy to fall into error. Maybe the psychology is clearer for other people. I hope it is.
p.s. I decided to start practicing not leaving comments because soon I won't be able to. I failed this time, but will do better. So, since this should be my last comment on one of your posts: I wish you a Happy Halloween, and a Merry Christmas!
The underlying maths has many other applications of course.
For example, suppose you're told that a facial recognition system has a 99% accuracy rate when comparing 2 head shots to decide whether those 2 head shots are the same person or not. We'll assume for simplicity that we have an equal rate of false positives (falsely saying it is the same person) and false negatives (falsely saying it is not the same person).
This sounds pretty good, and if you were to use it to ensure only the employees of a small company could enter its premises it would probably work well enough, at least if some allowance is made for the occasional mis-identification.
Now, we know that there's a 99/100 chance of correctly recording if 2 head shots are of the same person. But imagine using this to scan headshots taken (via CCTV) of members of the public as they walk down the streets of a busy city to check them against headshots of wanted criminals. Now the odds of getting things wrong become pertinent to your freedom to walk the streets unaccosted by the police.
How many comparisons would you need to do before the probability drops below 50%? The answer is 69. If you compare hundreds or thousands of people against the mugshots of wanted criminals you'll end up accusing lots of random civilians of being criminals and may well miss some of the wanted criminals along the way.
But suppose we raise the accuracy to 99.9%? The point at which the probability drops below 50% is 693. At 99.99% accuracy, the figure for dropping below 50% accuracy is 6,934. This application of facial recognition is going to need a very high accuracy to avoid lots of innocent people being flagged incorrectly as wanted criminals.
Now maybe there are ways around this by e.g. treating the initial match as a trigger for a deeper check of the person's identity to finally settle whether they're really the wanted criminal or not, but I'm sceptical of this kind of use of facial recognition because the maths indicate that an impressive level of pairwise-accuracy can nevertheless yield lots of error in such settings.
I was guessing over 365 to begin with. Lol. Interesting. Thanks for the shout out as well. :)
My very feeble old-man brain remembered the group size as 13, obviously wrong. But in any case, I always wondered how a small group could give me winning odds. I’ll probably stick with groups of 25 to get me closer to something like Vegas roulette house odds. Thanks for the explainer.
You remembered that the answer is a surprisingly small prime number whose second digit is a 3. That’s pretty good!
In my case, the leap from "does anyone in this crowd share a birthday? Want to bet?" to "does anyone in this crowd share MY birthday..." happens too fast for it to be a matter of simply substituting an easier question for a harder one. At first, I wanted to blame my listening skills, and then I wanted to blame my early training in memorizing the multiplcation tables ('what's 6 x 7? Quick! Wow, you need to be able to answer faster than that.'), but now I'm just not sure why it is so easy to fall into error. Maybe the psychology is clearer for other people. I hope it is.
p.s. I decided to start practicing not leaving comments because soon I won't be able to. I failed this time, but will do better. So, since this should be my last comment on one of your posts: I wish you a Happy Halloween, and a Merry Christmas!