Yes every bit of information that enables you eliminate the possibility that she has 2 girls born on Tuesday between 10-11 AM whilst not excluding the possibility that she has 2 boys born on Tuesday between 10-11 AM creates an ever so slightly imbalanced probability matrix. The rest of the boys and girls are symmetric but this imbalance here tips the scales a tiny tiny amount.
While the probability doesn’t work out the same, it’s fundamentally the same type of confusion as the Monty Hall problem - there’s more information revealed than you would think based on a cursory look at the problem. And the specific way the information is revealed is vital to understanding the probability
It would generally be 28 without some of the additional information but the other restrictions reduce the search space to one divisible by 27.
A simpler, classic example is “Mary has two children, at least one of which is a boy. What is the probability the other child is a boy?” The resulting probability is out of 3, not out of 4.
Ninja’d but also yes, the probability changes as more information is provided, e.g. hour of the day
That’s why they suck and why I wish their writers a violent bout of uncontrollable diarrhea. It’s all about verbal BS about exactly HOW they phrased their statement and how you handle the ambiguity.
In this one, there’s nothing that says or implies that the day of the week that the boy was born on has any bearing whatsoever on whether the other child is a girl.
That’s because the OP didn’t give the correct description of the problem. The issue isn’t that the writers are using verbal BS, it’s that mathmatical constructs don’t easily translate to conversational English. It’s much easier to describe these questions with mathmatical notation.
It may be that 51-52% of all people are male but that doesn’t necessarily mean that 51-52% of all births are male.
But maybe that’s the case?
It is always easier for me on these probability things if I can test it with a simulation, so I wrote a very simple program.
I get the 51.8% answer by simulation, but I have to be very careful to set the counting criteria to exactly follow the specifics in the questions.
# How many mothers
size=10000000
# For each mother generate two random births on two random days
flips <- data.frame(
flip1 = sample(1:2, size, replace=TRUE),
day1 = sample(c("SUN", "MON", "TUE", "WED", "THU",
"FRI", "SAT"), size, replace=TRUE),
flip2 = sample(1:2, size, replace=TRUE),
day2 = sample(c("SUN", "MON", "TUE", "WED", "THU",
"FRI", "SAT"), size, replace=TRUE)
)
# Save all the first born boys from Tuesday
tue1 <- flips[(flips$flip1==1 & flips$day1 == "TUE"),]
# Save all the second born boys from Tuesday, unless
# they have an older brother also born on Tuesday
tue2 <- flips[(flips$flip2==1 & flips$day2 == "TUE") &
!(flips$flip1==1 & flips$day1 == "TUE"),]
# merge the two datasets
tueboys <- rbind(tue1, tue2)
# save each mother who had a girl
girls <- tueboys[(tueboys$flip1==2 | tueboys$flip2==2),]
# compute the proportion of girls
nrow(girls)/nrow(tueboys)
Each mother has two children (or flips two coins, whatever). Each child is randomly born on one of the seven days.
I then select only those mothers who have a boy born on a Tuesday, and from those mothers I calculate how many’s other child is a girl.
The answer I get is 0.5187391. I can repeat this, as many times as I want because it takes a fraction of a second to run, but 51.8% appears to be the answer.
I understand the Monty Hall problem as adding new information. This one seems to me to be something of not using one random variable (sex), but two random variables (sex and day), which causes the odds to behave differently than one might expect.
Which is one perfectly valid way of setting up the problem, consistent with the text in the OP.
But it isn’t the only valid way of setting up the problem consistent with the text in the OP.
We can’t answer any question about the probability unless we know why she said what she did. The text the OP quoted doesn’t tell us that. Therefore, it is impossible to calculate an answer to the problem.
Yep. People get confused, because despite the label “probability problem”, the difficulty in finding the solution is actually about interpreting the words and not the mathematical computation. Some people may have trouble with doing the math anyway, but it’s the wording that eases or blocks comprehension. Because may people are insecure about innumeracy, they often blame their lack of understanding on that, instead of the poorly formed problem itself.
Whether the wording is ambiguous intentionally or accidentally is a separate issue.
People are discussing two different problems. The original poster gave us the link to an explanation of a problem, but when he quoted the problem he left out the “at least” –
““Mary has 2 children. She tells you that one is a boy born on a Tuesday. What’s the probability the other child is a girl?”
I bet the answer to that one is 50%.
To be clear, it’s the singular answer that’s impossible. A complete solution will have all possible answers. It’s like asking “what’s the square root of four?”.
I used to bug my high school math teachers by answering ambiguous questions with malicious interpretations. And then put the solution they expected in the margin.
Even saying “the other child” is a poor formulation, because there might not be a well-defined “the other child”. If both are boys born on Tuesday, which one is the “other” one?
Speaking as a math teacher, I’d welcome the challenge to improve the wording on my problems.
Exactly. The ambiguity of the problem leads to multiple ways it could be interpreted and solved.
For example, my first thought was that the day of the week isn’t pertinent, because based on what’s said, it’s no more relevant that had the question said “Mary has 2 children. She tells you that one is a boy with brown hair. What’s the probability the other child is a girl?
That said, we know that one child is a boy. So the other child is an independent event that has the same 1/2 chance of being a girl that any other birth has. So the answer is 1/2.
The Classic Boy/Girl paradox has 4 possible combinations for two kids:
- BB
- GG
- GB
- BG
If you know “(At least) One of them is a Boy” the probability that the other is a girl is 66.6%, as the information lets you eliminate the GG set, and 2/3 = 66.6% of the remaining sets have a girl.
If you know “The first one is a Boy”, the probability that the other is a girl is 50%, as the information lets you eliminate the GG and GB set, and 1/2 =50% of the remaining sets have a girl.
If you then introduce days of the week as a variable, your initial 4 possible combinations above now have 49 variations each (7 potential days of the week for the first kid multiplied by 7 potential days of the week for the second kid), 196 combinations total.
If you know “(At least) One of them is a Boy born on Tuesday”, the probability that the other is a girl is 51.8%, as:
- all 49 combinations of the GG set can be eliminated right off the bat, leaving 0,
- the BG set gets pared down to just 7 valid combinations for each day of the week for the girl
- the GB set is similarly reduced to 7 combinations
- the BB set is reduced to 13 combinations:
- B Tuesday for the first kid x 7 potential days of the week for the second kid, plus
- B Tuesday for the second kid x 6 potential days of the week (not 7 as the B-Tue, B-Tue set was already counted above) for the first kid.
So from the initial 196 combinations we have 13+7+7=27 valid combinations remaining, of which 14 have girls. 14/27 =51.8%
However, If you know “The first one is a Boy born on Tuesday”, the probability that the other is a girl is back to 50% , as you would have 7 valid combinations for each of the BG and BB sets
This is where tricky phrasing is used to make it not an independent event. See the difference in:
- A mother has at least one boy. If she has another child, what is the chance that child will be a girl.
- A mother has two children and one is a boy, what is the chance the other child is a girl.
It is. Typically quoted as 105 males born for every 100 females. Evens out over the years and by “old age” females predominate.
It’s unfortunate that the text explanation does not use the same formatting as in the video. Because when the combinations are laid out in a grid, as shown at 4:30 in the video, the hole were the other Boy Boy option is missing, makes the result easier to see.
Oh, I get it once the truth tables (or whatever they’re called for probabilistic problems) are laid out and all that.
I just get a bit of my high school and college math frustration and anger back when I read stuff like that, because it’s more about the wording and how you interpret the wording, than it’s about actually understanding the math itself.
So how does the following scenario fit in?
I deal down two cards out of a throughly mixed deck that is half G and half B.
Sight unseen each is 50/50 odds G or B.
I uncover one and see it is B. Tell you. You now know one of these two cards is a B. I then ask you what are the odds the remaining card is a G?
It is the same circumstance as phrased
I may be dense but I am not getting how the independence of what the second card is is contingent on my knowledge of the first card’s identity?
The way to do it to make it fit is
You draw 2 cards, look at them both: if they are both G you say nothing, replace them, draw 2 more cards, look at them both again, if they’re still GG you replace, draw again until you get a combination that is not GG.
Only then do you say “I have at least one B”
Looking at one card and stating its value of course the other card is 50-50
The intuitive answers are based on a problem setup like you describe, where one card (or child or whatever) is chosen randomly, and then we’re told information about that one, specific, chosen card (or child). These answers are intuitive because that’s the most common way that we gain information like this.
The non-intuitive answers are based on getting the information in other ways, such as someone dealing out two cards each from a bunch of decks, looking at all of the cards, and then eliminating all card pairs that do not meet some pre-decided condition. These answers are non-intuitive because that’s a very uncommon way for us to gain information like that.
The real problem is when people who don’t fully understand the problem don’t describe it fully, and then insist that the information must have been obtained in an uncommon way, instead of in the intuitive way. One sign of this happening is when the questioner uses phrases like “the other card”, which are only meaningful if you did it the intuitive way (which is why they’re using those phrases, because their own intuition works that way, too).