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

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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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