watches the people with basic math skills fight to the death over the answer
If you really wanna see a bloodbath, watch this:
You know that a couple has two children. You go to the couple’s house and one of their children, a young boy, opens the door. What is the probability that the couple’s other child is a girl?
Oops, I changed it to a more unintuitive one right after you replied! In my original comment, I said “you flip two coins, and you only know that at least one of them landed on heads. What is the probability that both landed on heads?”
And… No! Conditional probability strikes again! When you flipped those coins, the four possible outcomes were TT, TH, HT, HH
When you found out that at least one coin landed on heads, all you did was rule out TT. Now the possibilities are HT, TH, and HH. There’s actually only a 1/3 chance that both are heads! If I had specified that one particular coin landed on heads, then it would be 50%
This is basically Monty Hall right? The other child is a girl with 2/3 probability, because the first one being a boy eliminates the case where both children are girls, leaving three total cases, in two of which the other child is a girl (BG, GB, BB).
It’s somewhat ambiguous!
On the one hand, you might be right. This could be akin to flipping two coins and saying that at least one is heads. You’ve only eliminated GG, so BG, GB, and BB are all possible, so there’s only a 1/3 chance that both children are boys.
On the other hand, you could say this is akin to flipping two coins and saying that the one on the left (or the one who opened the door) is heads. In that case, you haven’t just ruled out GG, you’ve ruled out GB. Conditional probability is witchcraft
No, because knowing the first child is a boy doesn’t tell you any information about the second child.
Three doors, Girl Girl Boy
You select a door and Monty opens a door to show a Girl. You had higher odds of picking a girl door to start (2/3). So switching gives you better odds at changing to the door with the Boy because you probably picked a Girl door.
Here the child being a boy doesn’t matter and the other child can be either.
It’s 50/50 assuming genders are 50/50.
Well not really, right? BG and GB are the same scenario here, so it’s a 50/50 chance.
Even if, say, the eldest child always opened the door, it’d still be a 50/50 chance, as the eldest child being a boy eliminates the possibility of GB, leaving either BG or BB.
Cheeky bastard.
It is 50-50, though. The remaining possible states are BG and BB. Both are equally likely. Any further inference is narrative… not statistics.
The classic example of this is flipping 100 coins. If you get heads 99 times in a row… the last coin is still 50-50. Yes, it is obscenely unlikely to get heads 100 times in a row. But it’s already obscenely unlikely to get heads 99 times in a row. And it is obscenely unlikely to alternate perfectly between heads and tails. And it is obscenely unlikely to get a binary pattern spelling out the alphabet. And it is obscenely unlikely to get… literally any pattern.
Every pattern is equally unlikely, with a fair coin. We see 99 heads in a row versus 1 tails at the end, and think it narrowly averted the least-probable outcome. But only because we lump together all sequences with exactly one tails. That’s one hundred different patterns. 1-99 is not the same as 99-1. We just treat them the same because we fixate on uniformity.
Compare a non-binary choice: a ten-sided die. Thirty 1s in a row is about as unlikely as 100 heads in a row. But 1 1 1… 2 is the same as 1 1 1… 3. Getting the first 29 is pretty damn unlikely. One chance in a hundred million trillion. But the final die can land on any number 1-10. Nine of them upset the pattern our ape brains want. Wanting it doesn’t make it any more likely. Or any less likely.
It would be identically unlikely for a 10-sided die to count from 1 to 10, three times in a row. All the faces appear equally. But swap any two events and suddenly it doesn’t count. No pun intended.
If this couple had eight children, for some god-forsaken reason, and you saw seven boys, the eighth kid being another boy is not less likely for it. The possibility space has already been reduced to two possibilities out of… well nine, I suppose, if order doesn’t matter. They could have 0-8 boys. They have at least 7. The only field that says the last kid’s not a coin toss is genetics, and they say this guy’s chromosome game is strong.
You’re right, but it’s not a subversion of the Gambler’s Fallacy, it’s a subversion of conditional probability. A classic example is that I have two kids, and at least one of them is a boy. What is the probability that I have two boys?
The intuitive answer is 50%, because one kid’s sex doesn’t affect the other. But when I told you that I have two kids, there were four possibilities: GG, GB, BG, or BB. When I told you that at least one of them is a boy, all I did was take away the GG option. That means there’s only a 1 in 3 chance that I have two boys.
But by having one child answer the door, I change it yet again–now we know the sex of a particular child. We know that the child who opened the door is a boy. This is now akin to saying “I have two children, and the eldest is a boy. What is the possibility that I have two boys?” It’s a sneaky nerd snipe, because it targets specifically people who know enough about statistics to know what conditional probability is. It’s also a dangerous nerd snipe, because it’s entirely possible that my reasoning is wrong!
i hate it when ppl do nb erasure for their stupid math text problems. use anything else pls
Two more for funsies! I flipped two coins. At least one of them landed on heads. What is the probability that both landed on heads? (Note: this is what my comment originally said before I edited it)
I have two children. At least one of them is a boy born on a Tuesday. What is the probability that I have two boys?
Different compilers have robbed me of all trust in order-of-operations. If there’s any possibility of ambiguity - it’s going in parentheses. If something’s fucky and I can’t tell where, well, better parenthesize my equations, just in case.
This is best practice since there is no standard order of operations across languages. It’s an easy place for bugs to sneak in, and it takes a non-insignificant amount of time to debug.
there is no standard order of operations across languages
Yes there is. The rules of Maths are universal.
It’s an easy place for bugs to sneak in
But that’s because of programmers not checking the rules of Maths first.
This is the way. It’s an intentionally ambiguously written problem to cause this issue depending on how and where you learned order of operations to cause a fight.
intentionally ambiguously written
#MathsIsNeverAmbiguous
learned order of operations to cause a fight
The order of operations are the same everywhere. The fights arise from people who don’t remember them.
Please see this section of Wikipedia on the order of operations.
The “math” itself might not be ambiguous, but how we write it down absolutely can be. This is why you don’t see actual mathematicians arguing over which one of these calculators is correct - it is not either calculator being wrong, it is a poorly constructed equation.
As for order of operations, they are “meant to be” the same everywhere, but they are taught differently. US - PEMDAS vs UK - BODMAS (notice division and multiplication swapped places). Now, they will say they are both given equal priority, but you can’t actually do all of the multiplication and division at one time. Some are taught to simply work left to right, while others are taught to do multiplication first; but we are all taught to use parentheses correctly to eliminate ambiguity.
Different compilers
Different programmers.
it’s going in parentheses
Unfortunately some places don’t care where you’ve put brackets, they’ll just go ahead and change it anyway. Welcome to my quest to educate.
That’s the same ambiguity, numbnuts. Your added parentheses do nothing. If you wanted to express the value 8 over the value 2*(1+3), you should write 8/(2*(1+3)). That is how you eliminate other valid interpretations.
As illustration of why there are competing valid interpretations: what human being is going to read “8/2 * (1+3)” as anything but 4*4? Those spaces create semantic separation. But obviously most calculators don’t have a spacebar, any more than they have to ability to draw a big horizontal line and place 2(1+3) underneath it. Ambiguous syntax for expressing mathematics is not some foundation-shaking contradiction. It’s a consequence of limitations in how we express even the most concrete ideas.
“The rules of math” you keep spamming about are not mathematical proofs - they’re arbitrary decisions made by individuals and organizations. In many cases the opposite choice would be equally sensible. Unlike the innate equivalence of multiplication and division, where dividing by two and multiplying by half are interchangeable. Same with addition and subtraction.
Do you want to argue that 8 - (2) + (1+3) should be 2?
Your added parentheses do nothing
So you’re saying Brackets aren’t first in order of operations? What do you think brackets are for?
If you wanted to express the value 8 over the value 2*(1+3), you should write 8/(2*(1+3))
or, more correctly 8/2(1+3), as per the rules of Maths (we never write unnecessary brackets).
That is how you eliminate other valid interpretations
There aren’t any other valid interpretations. #MathsIsNeverAmbiguous
what human being is going to read “8/2 * (1+3)” as anything but 4*4
Yes, that’s right, but 8/2x(1+3) isn’t the same as 8/2(1+3). That’s the mistake that a lot of people make - disobeying The Distributive Law.
Those spaces
…have no meaning in Maths. The thing that separates the Terms, in your example, is the multiply. i.e. an operator.
most calculators don’t have a spacebar
…because it’s literally meaningless in Maths.
any more than they have to ability to draw a big horizontal line and place 2(1+3) underneath it
Some of them can actually.
“The rules of math” you keep spamming about are not mathematical proofs
You should’ve read further on then. Here’s the proof.
they’re arbitrary decisions made by individuals
No, they’re a natural consequence of the way we have defined operators. e.g. 2x3=2+2+2, therefore we have to do multiplication before addition.
In many cases the opposite choice would be equally sensible
2+2x3=2+6=8 the correct answer, but if I do addition first…
2+2x3=4x3=12, which is the wrong answer. How is getting the wrong answer “equally sensible” as getting the right answer?
Do you want to argue that 8 - (2) + (1+3) should be 2?
No, why would I do that? 8-(2+1+3) does equal 2 though.
There’s quite a few calculators that get this wrong. In college, I found out that Casio calculators do things the right way, are affordable, and readily available. I stuck with it through the rest of my classes.
Casio does a wonderful job, and it’s a shame they aren’t more standard in American schooling. Texas Instruments costs more of the same jobs, and is mandatory for certain systems or tests. You need to pay like $40 for a calculator that hasn’t changed much if at all from the 1990’s.
Meanwhile I have a Casio fx-115ES Plus and it does everything that one did, plus some nice quality of life features, for less money.
TI did the same thing Quark and Adobe did later on – got dominance in their markets, killed off their competition, and then sat back and rested on their laurels thinking they were untouchable
EDIT: although in part, we should thank TI for one thing – if they hadn’t monopolized the calculator market, Commodore would’ve gone into calculators instead of computers
https://en.wikipedia.org/wiki/TI-99/4A
It was a huge failure, but they tried.
My Casio calculators get this wrong, even the newer ones. BTW the correct answer is 16, right?
- 16 is the right answer if you use PEMDAS only:
(8 ÷ 2) × (2 + 2) - 1 is the right answer if you use implicit/explicit with PEMDAS:
8 ÷ (2 × (2 + 2)) - both are correct answers (as in if you don’t put in extra parentheses to reduce ambiguity, you should expect expect either answer)
- this is also one of the reasons why postfix and prefix notations have an advantage over infix notation
- postfix (HP, RPN, Forth):
2 2 + 8 2 ÷ × . - prefix (Lisp):
(× (÷ 8 2) (+ 2 2))
- postfix (HP, RPN, Forth):
prefix notation doesn’t need parentheses either though, at least in this case. lisp uses them for readability and to get multiple arity operators. infix doesn’t have any ambiguity either if you parenthesize all operations like that.
16 is the right answer if you use PEMDAS only: (8 ÷ 2) × (2 + 2)
You added brackets and changed the answer. 2(2+2) is a single term, and if you break it up then you change the answer (because now the (2+2) is in the numerator instead of in the denominator).
1 is the right answer
The only right answer
both are correct answers
Nope, 1 is the only correct answer.
this is also one of the reasons why postfix and prefix notations have an advantage over infix notation
Except they don’t. This isn’t a notation problem, it’s a people don’t remember the rules of Maths problem.
Write it out on paper, with numerator above the denominator. You’d have to write 8(2+2), 2(2+2) can only be written if both are in the denominator.
Depends on the system you use. Most common system worldwide and in the academic circles (the oldest of the two) has 1 as the answer.
the correct answer is 16, right?
In some countries we’re taught to treat implicit multiplications as a block, as if it was surrounded by parenthesis. Not sure what exactly this convention is called, but afaic this shit was never ambiguous here. It is a convention thing, there is no right or wrong as the convention needs to be given first. It is like arguing the spelling of color vs colour.
This is exactly right. It’s not a law of maths in the way that 1+1=2 is a law. It’s a convention of notation.
The vast majority of the time, mathematicians use implicit multiplication (aka multiplication indicated by juxtaposition) at a higher priority than division. This makes sense when you consider something like 1/2x. It’s an extremely common thing to want to write, and it would be a pain in the arse to have to write brackets there every single time. So 1/2x is universally interpreted as 1/(2x), and not (1/2)x, which would be x/2.
The same logic is what’s used here when people arrive at an answer of 1.
If you were to survey a bunch of mathematicians—and I mean people doing academic research in maths, not primary school teachers—you would find the vast majority of them would get to 1. However, you would first have to give a way to do that survey such that they don’t realise the reason they’re being surveyed, because if they realise it’s over a question like this they’ll probably end up saying “it’s deliberately ambiguous in an attempt to start arguments”.
The real answer is that anyone who deals with math a lot would never write it this way, but use fractions instead
So are you suggesting that Richard Feynman didn’t “deal with maths a lot”, then? Because there definitely exist examples where he worked within the limitations of the medium he was writing in (namely: writing in places where using bar fractions was not an option) and used juxtaposition for multiplication bound more tightly than division.
Here’s another example, from an advanced mathematics textbook:
Both show the use of juxtaposition taking precedence over division.
I should note that these screenshots are both taken from this video where you can see them with greater context and discussion on the subject.
The real answer is that anyone who deals with math a lot would never write it this way
Yes, they would - it’s the standard way to write a factorised term.
but use fractions instead
Fractions and division aren’t the same thing.
So 1/2x is universally interpreted as 1/(2x), and not (1/2)x, which would be x/2.
Sorry but both my phone calculator and TI-84 calculate 1/2X to be the same thing as X/2. It’s simply evaluating the equation left to right since multiplication and division have equal priorities.
X = 5
Y = 1/2X => (1/2) * X => X/2
Y = 2.5
If you want to see Y = 0.1 you must explicitly add parentheses around the 2X.
Before this thread I have never heard of implicit operations having higher priority than explicit operations, which honestly sounds like 100% bogus anyway.
You are saying that an implied operation has higher priority than one which I am defining as part of the equation with an operator? Bogus. I don’t buy it. Seriously when was this decided?
I am no mathematics expert, but I have taken up to calc 2 and differential equations and never heard this “rule” before.
I can say that this is a common thing in engineering. Pretty much everyone I know would treat 1/2x as 1/(2x).
Which does make it a pain when punched into calculators to remember the way we write it is not necessarily the right way to enter it. So when put into matlab or calculators or what have you the number of brackets can become ridiculous.
Sorry but both my phone calculator and TI-84 calculate 1/2X
…and they’re both wrong, because they are disobeying the order of operations rules. Almost all e-calculators are wrong, whereas almost all physical calculators do it correctly (the notable exception being Texas Instruments).
You are saying that an implied operation has higher priority than one which I am defining as part of the equation with an operator? Bogus. I don’t buy it. Seriously when was this decided?
The rules of Terms and The Distributive Law, somewhere between 100-400 years ago, as per Maths textbooks of any age. Operators separate terms.
I am no mathematics expert… never heard this “rule” before.
I’m a High School Maths teacher/tutor, and have taught it many times.
It’s not a law of maths in the way that 1+1=2 is a law
Yes it is, literally! The Distributive Law, and Terms. Also 1+1=2 isn’t a Law, but a definition.
So 1/2x is universally interpreted as 1/(2x)
Correct, Terms - ab=(axb).
people doing academic research in maths, not primary school teachers
Don’t ask either - this is actually taught in Year 7.
if they realise it’s over a question like this they’ll probably end up saying “it’s deliberately ambiguous in an attempt to start arguments”
The university people, who’ve forgotten the rules of Maths, certainly say that, but I doubt Primary School teachers would say that - they teach the first stage of order of operations, without coefficients, then high school teachers teach how to do brackets with coefficients (The Distributive Law).
BEDMAS: Bracket - Exponent - Divide - Multiply - Add - Subtract
PEMDAS: Parenthesis - Exponent - Multiply - Divide - Add - Subtract
Firstly, don’t forget exponents come before multiply/divide. More importantly, neither defines wether implied multiplication is a multiply/divide operation or a bracketed operation.
Exponents should be the first thing right? Or are we talking the brackets in exponents…
afair, multiplication was always before division, also as addition was before subtraction
It’s BE(D=M)(A=S). Different places have slightly different acronyms - B for bracket vs P for parenthesis, for example.
But multiplication and division are whichever comes first right to left in the expression, and likewise with subtraction.
Although implicit multiplication is often treated as binding tighter than explicit. 1/2x is usually interpreted as 1/(2x), not (1/2)x.
Multiplication VS division doesn’t matter just like order of addition and subtraction doesn’t matter… You can do either and get same results.
Edit : the order matters as proven below, hence is important
I think when a number or variable is adjacent a bracket or parenthesis then it’s distribution to the terms within should always take place before any other multiplication or division outside of it. I think there is a clear right answer and it’s 1.
No there is no clear right answer because it is ambiguous. You would never seen it written that way.
Does it mean A÷[(B)©] or A÷B*C
It means
A ÷ B(C) which is equivalent to A ÷ (B*C)
I literally just explained this. The Parenthesis takes priority over multiplication and division outright.
Maybe
B*C = B(C)
But
A ÷ B(C) =! A ÷ B * C
Not sure what exactly this convention is called
It’s The Distributive Law
It is a convention thing, there is no right or wrong
No, it’s an actual rule, and 1 is the only correct answer here - if you got 16 then you didn’t obey the rule.
Please Excuse My Dear Aunt Sally, she downloaded a shitty ad-infested calculator from the Google Play store.
