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This question already has an answer here:

Discrete Probability: Four dice are thrown, what's the probability that...

So I made a post and there were a lot of incorrect answers, and two people were able to provide the correct answers, but I cannot understand why or how they got it.

So for this particular question: A dice is thrown four times, what's the probability "four" is the highest number thrown?

Okay, so what I understand is the sample space is $6^4$. Now we need to find the probability that $4$ is the highest number thrown. So that eliminates 2 other possible outcomes, 5 and 6. So we have 4 possible outcomes now. So $\frac{4^{4}}{6^{4}}$. But now what I cannot understand is why on earth do we need to subtract the number of possibilities where none of the numbers fall higher than 3? Like why does it matter? What if 4 is rolled all the time? Who cares?

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marked as duplicate by David K, GNUSupporter 8964民主女神 地下教會, Leucippus, Lord Shark the Unknown, José Carlos Santos Feb 17 at 9:33

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  • $\begingroup$ When you consider the event that the maximum is less than or equal to $4$, it includes the case where it is less than $4$, which you do not want - you want the maximum equals to $4$. So you subtract the unwanted case, getting the relative complement of the event. $\endgroup$ – BGM Feb 16 at 19:35
  • $\begingroup$ “4 is the highest “ also means that a 4 had to happen. $\endgroup$ – Randall Feb 16 at 19:47
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There are $4^4$ ways in which we can roll either $1$, $2$, $3$ or $4$ in $4$ rolls of a die. But this includes the situations in which no $4$ is rolled. In these cases $4$ is not the maximum number. The number of ways in which we can roll the numbers $1$, $2$ or $3$ exclusively is $3^4$. So the probability of rolling $4$ as the highest number is $4^4-3^4$ divided by the sample space - $6^4$. $$\frac{4^4-3^4}{6^4}=\frac{175}{1296}$$

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The $4^4$ is the number of ways to get all $4$ dice rolls be between $1$ and $4$ (inclusive). However, this also includes the $3^4$ cases where all the dice rolls were between $1$ and $3$. We do not want to include these cases (since if all dice rolls are between $1$ and $3$, then the maximum roll cannot be $4$, because $4$ isn't rolled!). Therefore, we subtract it off.

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