The Hidden Power of Information: How Small Details Can Change Everything

A Probability Puzzle That Reveals Why What You Say (and Don’t Say) Matters

This is The Curious Mind, by Álvaro Muñiz: a newsletter where you will learn about technical topics in an easy way, from decision-making to personal finance.

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When sharing information, the way we present it can dramatically affect outcomes.

Whether in puzzles, interviews, or everyday life, the key is to be careful about what we disclose. Information carries power—not only in what it explicitly conveys but also in what it implies. Being strategic with the details you provide can often make all the difference.

Today we’ll see how subtle details can change everything.

Source: ChatGPT

Boys & Girls

Let’s start, as we like to do, with a puzzle.

Ms. Jackson and Ms. Parker are both mothers of two children.¹

They work at a company that is hosting a dinner for working mothers with at least one son (male). Attendees may bring one child if they choose.

• Ms. Jackson is invited and attends on her own.

• Ms. Parker is also invited and attends with her son (male).

Assume that each child is equally likely to be a boy or a girl.

What do you think?

I am curious to know what are your first thoughts: please answer the poll above and feel free to write your argument in the comments below.

Information and Updating our Believes

Obvious Information

This is a probabilistic question in which we are given some prior information.

If we were not told anything about either Ms. Jackson or Ms. Parker, each of them could have:

1. 2 boys.

2. 2 girls.

3. An older boy and a younger girl.

4. A younger boy and an older girl.

All these possibilities are equally likely, so each of them has probability 25%. In particular, the probability that Ms. Jackson has two boys would be the same as the probability that Ms. Parker has two boys, both being 25%.

Instead, we are given relevant information about both mothers, and this should update our beliefs. We are told that both mothers are invited to the dinner, which rules out the possibility that they have no boys.

That is, the fact that both mothers were invited to the dinner means that each of them could either have:

1. 2 boys.

2. 2 girls

3. An older boy and a younger girl.

4. A younger boy and an older girl.

Therefore, when a mother is invited to the dinner, the probability she has two boys is now 33%. ²

So, are both Ms. Jackson and Ms. Parker equally likely to have two boys, the probability being 33%?

Not-So-Obvious Information

It seems we have concluded that they should both be equally likely to have two boys.

Indeed, this is what I first thought when I encountered a similar version of this problem. But then I thought:

Wait a second. Why are they telling me that one of them attended with her son, while the other didn’t. Is this useless information that is only trying to confuse me, or is it actually relevant?

At first, it seemed to me like this extra piece of information was redundant. Here is what I thought:

• Ms. Jackson is invited, so she has at least one son. She comes to the dinner alone. Thanks for the info, but all I can say after that is the same: Ms. Jackson has at least one son.

• Ms. Parker is invited, so she has at least one son. She comes to the dinner with her son. Thanks for the info, but all I can say after that is pretty much the same: Ms. Parker has at least one son (and, in case she has two, she likes the one she brought a bit more?).

It is true that Ms. Jackson coming to the dinner alone doesn’t reveal any useful information. However, here is a very subtle misconception in the argument about Ms. Parker:

Ms. Parker comes to the dinner with her son. Therefore, that particular child is a boy.

Why does this make a difference?

If that particular child is a boy, then the probability that both her children are boys is just the probability that her other child is a boy. This one right here is already a boy—it is just a matter of the other child.

Since we assume that every child is equally likely to be a boy or a girl, the probability that Ms. Parker has two boys is 50%.

So, while Ms. Jackson is only 33% likely to have two boys, Ms. Parker’s probability is considerably higher, having two boys half of the time.

Information in Real-Life

I had to think about the Boys & Girls solution (not to mention the puzzle itself) for a while before fully understanding what was going on.

The key is that knowing that Ms. Jackson was invited to the dinner doesn't single out any of her children: we know that she has at least one boy, but her children are still undistinguishable. On the other hand, when we see the boy next to Ms. Parker at the dinner, her children are no longer undistinguishable: there is one that is here, and other that is not.

This gives us an important takeaway:

When you need to give out information, you must be very careful about how much you give—convey only what is strictly necessary. There might be subtle ways in which extra information can be used.

In particular, when discussing about things like occurrences or possessions, never single out any particular occurrence or possession.

Here are some examples:

• If you are asked at an interview whether you have been fired before—and assuming you need to tell the truth—, never say anything like "I was fired in my first/previous/CompanyX job". Convey the information that you have been fired, but without singling out any particular instance.

• If you are trying to get a mortgage and are asked whether you have any other loans, do not say “I have a loan with bank X”. Convey the truthful information that you do have another loan, but without giving the information of which particular bank.³

• If you are playing a cards game and need to reveal whether you have an Ace, do not point out to a card and say it is an Ace (even if you shuffle your cards later). Admit that you do have an Ace, but do not single out any of your cards.

Information is powerful not only in what it explicitly states but also in what it implicitly reveals. The key is to be precise and strategic in the details you disclose. When in doubt, the safest thing is to convey as little as possible.

So, next time you answer a question, think:

Am I giving exactly the information required, or am I also handing over insights that will be used in ways I didn’t intend?

In case you missed it

• From Profit to Utility: Approaching Risk in Financial Decisions

• The Casino Always Wins: The Math Behind Roulette

• Selling your iPhone in 14 Days: How to Choose the Best Offer Using Math

What do you want next?

1. For Spanish readers: children can be either male or female. .

2. If you are wondering why it is 33%, note that the fact that a mother is invited to the dinner doesn’t make either of the cases 1 (2 boys), 3 (an older boy and a younger girl) or 4 (a younger boy and an older girl) more likely: she will be invited regardless of whether she has 1 or 2 boys, and regardless of their age. So each of the three cases is still equally likely, each with a probability of 33%.

Another option is to use Bayes’ theorem, as we learned a couple weeks ago (give it a try!).

3. Of course, you might be later asked about which bank, in which case you will need to give out that information too. But not yet!