Is guessing a number out of 10 equally as "impressive" as picking a name out of 10 in a hat finding out it's your ex's name?

Question

This is a question that depending on how it's framed, was making me question whether or not these events are equally "impressive".

If I guessed a number out of 10 before randomly generating it, getting an equal match has a probability of 1/10. Now, picking a name out of a hat and having it match your ex's name is also 1/10, assuming that that was the only name that was your ex's name. It seems that they're equally impressive.

But what if the other names in the hat had some sort of meaning. What if one of the names was your mother's and another was your sister's. Now, the chance of picking a meaningful name out of the hat is 3/10. Whereas the chance of picking a meaningful guess out of 10 which in this case would only make sense to be meaningful if it's a correct guess will remain 1/10.

When framed in this way, correctly guessing a number seems more rare.

Which is more impressive?

Answer 1

Your apriori knowledge in the two guesses is very different. In the integer number guess you know you only have 10 choices: 1-10.

In the name guess, 10 people put their name into a hat. As a participant, If I know the names of all the people in the hat, then the number guess and the integer guess are equivalent.

Your experiment assumes you know there are 10 names in the hat but you have no idea what the names are. So if your ex name was Zoe, you would have to include the probability of getting the name Zoe out of all the possible names on the planet.

Answer 2

Part of the surprise has to do with the implicit statistical assumptions and part with the importance attributed to the choice.

Statistical assumptions (uncertainty reduction)
When guessing a number out of 10, one usually assumes a uniform distribution, i.e., that any number can come with equal probability. This is not the only possible choice, but this is what most people expect, when asked to pick/guess a number out of 10.

The situation with one girlfriend's name is less clear - it depends on how the names were chosen, how many girlfriends one has had, how common are their names, etc. E.g., none of one's girlfriends names could be in a hat, so the surprise might be at the fact that the name is in the hat at all. If the name is Jane than picking it is less surprising than if it were Rekefet. If you've had many girlfriends, it is more probable that a name of one of them is in the hat - a Korean friend once cited me a saying, describing this situation: If I throw a stone from a mountain, I'll kill Kim, Lee or Park.

Now, if we know a priori that one of the names in the hat is the one's girlfirend's name, and there is only one such name in the hat, than we are in the same situation, as when guessing number out of 10. Hats and other similar devices are used to simulate the uniform probability distribution that I mentioned above - any item is equally likely to be drawn.

This kind of "suprise" is formalizable and can be characterized as the reduction of uncertainty or information gained when making a choice. In fact, in information theory textbooks uncertainty/information is often introduced precisely in this way - as the degree of surprise.

Psychological factor
Assuming that the probabilistic conditions are the same - i.e., there is one and only one girlfriend's name in the hat - drawing one's girlfreends name would be still more surprising than drawing a particular number. Indeed, if we replaced the names in the hat with numbers, the statistical conditions would be the same, but the surprise would be less... unless one attaches specific significance to certain numbers. E.g., a common superstition is to attribute importance to number 13, even when it is equally drawn from, e.g., 200 hotel suits or 50 places in the bus. Same can be said about a number that is the day of one's birth or some other significant date (e.g., if you are American drawing two numbers and getting 9 and 11.)

Just in the same way, one's girlfriends name is different from other names by the significance one attaches to it: the happy memories, or the difficult separation, or perhaps possibility of making-up.

In statistical hypothesis testing one usually uses procedures specifically designed to minimize the influence of the emotional or other kinds of importance attached to the results. Thus, the assumptions and expectations have to be stated in advance, to avoid attributing importance to the result post-hoc, i.e., after the choice/experiment has been carried out (and scientists often have a lot of emotional attachment to confirming their pet theory, proving that the years of work were not wasted, having something to publish/put in one's thesis or getting research funding.)

Answer 3

First of all the question would be what the set of possibilities is and whether you know what this set of possibilities is.

So if 10 numbers are randomly sampled from a set of all numbers and then you match a number that is literally infinitely more unlikely than if you did the same for a uniform set of numbers from 1-10.

Same with names. There are billions of names some are intrinsically given more often and thus more likely to be in the hat sampled from all names or in the set of your ex-girlfriends and other's are less so.

So whether the set is known or unknown to the person choosing makes a significant difference as to how surprised they are by the result.

So it could be 10x the same name so a probability of 1 to draw it, but if you expect a set of all possible names and a uniform random sampling then this would still feel impressive to you, that is before the other options are revealed.

Also the set of all possible numbers and names are not of equal size so technically the set of all names is limited the set of all numbers is unlimited, yet the set of names is huge while the practically used set of all numbers probably soft caps at 1000 after which people would rather name orders of magnitude than numbers (million, billion rather than 53,897), so despite being larger the practical range of numbers might be significantly smaller.

So if you had this "all possible X" in mind you would probably find names more impressive. Another aspect is that when you're dealing with numbers you might already be more inclined to activate the parts of your brain concerned with abstraction and math, meaning you'd more likely compute probabilities and whatnot while names wouldn't do that trick. If you're given 10 names you'd first have to count that they are 10 and from there you'd compute the probability so there would be further steps that you'd need to take making it more impressive because you're further removed from that level of abstraction.