Is an object confirmation formula possible to derive in order to predict the best number of guesses that should be made?

That would depend on how people select the numbers, but yes, the largest amount of selections would result in the highest accuracy.

Assuming people pick 1 option at random, the likelihood of at least 1 match would be:

1 - 24/25 \* 23/25 = 1 - 552/625 = 73/625 = 11.68%

However, if each person is allowed to pick 3 different options at random, then the likelihood of at least 1 match would be:

1 - 22/25 \* 19/25 = 1 - 418/625 = 207/625 = 33.12%

In other words, the likelihood of at least 1 match goes up the more selections you have.

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It can be useful to imagine the edge case: "what if each person has 25 different selections?" Well, then the accuracy would be 100% all the time. And you can imagine a sort of slider from this edge case to the following: "what if each person has 0 selections?", where the accuracy would be 0% all the time.

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Now, if the selections are not different, i.e. each person is allowed to repeat a selection, then you can imagine an even edgier case: "what if each person has an amount of selections that approaches infinity?" - clearly, in this case, you can see that the accuracy will also approach 100%.

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Hopefully I've understood your question correctly. If you have meant something else, then please tell me. If I have understood it correctly, then it's not really an analysis question as much as it is a probability theory question.

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