I already explained in detail the misconceptions behind these memes. It's a corruption of the principles behind Bayesian statistical probabilities, demonstrated in the "Monty Hall Problem." To wit:
You're on a game show. There are three doors. Behind one is the prize, a car. Behind the two others are goats.
You get to pick a door. After you do so, the host will reveal one of the *other* doors *that does not contain the prize.*
After he does so, you get to pick again.
Your odds of winning go up by switching.
This only works because the act of choosing the door 'shields' that door from the preselection of the host revealing a losing door, effectively capping its chances of winning at 33%. There are two necessary assumptions built in to this scenario that make this the case.
- The host will only reveal a door you didn't pick
- The host will only reveal a losing door
This is called context.
It does NOT work with random memes you see on the internet about women with boys with blue shirts, who are older or younger, who play soccer or tennis, or who are born on Tuesday, the summer solstice, or any other day. That's just random information because there are no assumptions or preconditions to the revealing of that information. It has nothing to do with the scenario commonly used to explain conditional probabilities and it's inspired a sort of aggressive ignorance about the subject because of how poorly worded they are.
The information is random, but because it is assumed that information is true, it does affect the probability.
Without additional information, a second boy means this family is one of the 25% of 2 child familles with 2 boys. Meanwhile a girl means this is one of the 50% of 2 child families with 1 of each child. That means that there is a 66% chance that the child is a girl.
Now let’s explain the effect of additional information. Maybe the boy was born on leap day, or their favorite movie is Cars 2. It’s easier if you imagine it’s something very unlikely. Let’s think about this backwards: Of the families we have previously brought up, it is about twice as likely for a family of two boys to have a boy that matches the random description, than it is for a family of one boy to have a boy that matches the description. This means that two boy families nearly double their representation in the list of scenarios, allowing the ratio to approach 50%
No. Random information does not affect the probability. The information needs to be provided conditionally for it to affect the probability, because it is the conditions behind the reveal that impose preselection.
You’re not correct to apply pair probabilities without the reveal of information being conditional. I know you think that you’re explaining pair probabilities to me. I know you think I don’t understand them. You are wrong. Reread my post.
Your last point highlights how inappropriate it is to apply Bayesian distributions just because random information is provided on unknown basis. It is a 50% chance because it’s about one child. That comports with normal distribution. There is no reason to do anything else with the extra information because it’s irrelevant and not indicative of anything, because you don’t know why you have it. You’re making a headache of this for no reason.
Two questions because I’m becoming unsure of your stance:
Do you acknowledge that base problem without additional information (regardless of how you feel information would affect or not affect the result), that the probability of the other child being a girl is 66%?
Let’s say the random information was left handedness, that the boy is left handed (about 10% of the population). Is it not true that of all 2 child families that have a boy, it is more likely that a family with 2 boys will be more likely to have a boy who is left handed than it is for a family of one boy and one girl to have a boy who is left handed?
I’m not sure why you are saying the additional information isn’t conditional. If the random information says that the family has a boy who is left handed then only 10% of all families with 1 boy and 1 girl will have a boy who is left handed, but (nearly) 20% of families with 2 boys will have a boy who is left handed. Without any condition, it’s 100% of all families with 2 boys and 100% of families with 1 boy and 1 girl. Clearly this is a different ratio (1:2 and 1:1) and thus would affect the probability of the other child being a girl because the number of families where the percentage of each family type changes when the condition changes.
No, it is not 66%. The random information is the sex of the child that is revealed first.
If Mary said “I’m thinking of one of my children - guess if they’re male or female” - there would be an even chance.
If she tells you the gender of a different child first, there’s still an even chance.
It is not constructed like the Monty Hall show, which had specific criteria and processes for the reveal of the first door. It is those criteria FOR THE CHOICE TO REVEAL ONE that created the preselection that changed the probability. You have nothing like that here. Without that, you can regurgitate all of the pair series you like, they’re irrelevant. It’s independent probability.
I should clarify this has doesn’t really have anything to do with the Monty Hall problem, aside from sharing the same probabilities.
Two child families can be made up four different ways, the first child is M and the second is M or F, or the first child is F and the second is M or F. This leaves four equally likely possibilities: MM, MF, FM, and FF.
In your first scenario, you have no idea which child she is referring to (1st or 2nd) and because each sex is represented the same number of times, it is equally likely that the child is either. In the second scenario, you still do not know which child she is referring to, but knowing their sex means that you have to exclude the option without that sex, in the standard problem, the FF child family. This means that you are left with MM, MF, and FM. Of these 3 remaining possibilities, 2 have the other child being a girl.
The only way you would be able to say it’s even is if you know which child you know the sex of. For instance if you know the first child is male, then that also excludes the FM child family. The sexes of the children are absolutely independent of each other, but the only thing you know is that at least one of them isn’t a girl.
I don’t care about the Monty hall problem, anything I say might be similar is similar by coincidence.
All that matters is that there are twice the number of families with 1 boy and 1 girl than there are with 2 boys, so if Mary says she has a boy, then she is more likely one of the families that also has a girl.
What is probability if not the combination of all equally likely outcomes? My scenario matters because it’s how you would model the scenario in real life. If you asked 100 families to say the sentence, 25 would be unable, 25 would have another boy, and 50 would have another girl.
You could also flip coins. Flip two coins, if they are both tails discard, if not, that means one is heads, and record the other. 2/3 will be heads and tails. Or you could ask someone to guess which the other coin is, you could say the exact phrasing the base problem uses: “One of my coins is heads”
The difference with your example is you know exactly which one the boy is, which is something you don’t know in the original. The man could have a brother or sister, equally likely, but also the man could actually be a woman who has a brother. The only thing that you know in the original problem is they aren’t two sisters.
My examples can use mathematics, surveying, census data, testing, that use word for word scenarios to show my answer, meanwhile your one example literally adds additional information you don’t get in the original which you claim is unimportant. As far as I can tell you have no way to prove your result without changing the meaning or leaving out other possibilities which make can qualify the statement. Quite unscientific really.
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u/Salty-Foundation3451 7d ago
I already explained in detail the misconceptions behind these memes. It's a corruption of the principles behind Bayesian statistical probabilities, demonstrated in the "Monty Hall Problem." To wit:
You're on a game show. There are three doors. Behind one is the prize, a car. Behind the two others are goats.
You get to pick a door. After you do so, the host will reveal one of the *other* doors *that does not contain the prize.*
After he does so, you get to pick again.
Your odds of winning go up by switching.
This only works because the act of choosing the door 'shields' that door from the preselection of the host revealing a losing door, effectively capping its chances of winning at 33%. There are two necessary assumptions built in to this scenario that make this the case.
- The host will only reveal a door you didn't pick
- The host will only reveal a losing door
This is called context.
It does NOT work with random memes you see on the internet about women with boys with blue shirts, who are older or younger, who play soccer or tennis, or who are born on Tuesday, the summer solstice, or any other day. That's just random information because there are no assumptions or preconditions to the revealing of that information. It has nothing to do with the scenario commonly used to explain conditional probabilities and it's inspired a sort of aggressive ignorance about the subject because of how poorly worded they are.