Say you flip a coin 5 times, and are asked two questions:

1. what's the probability you get more Heads than Tails?

2. what's the probability you get more Tails than Heads?

the answer to those two questions must be the same, because, since it's equally likely you get Heads or Tails on each toss, there's no logical reason why the answer to question 1 should be lower or higher than the answer to question 2.

But the situations in 1 and 2 are the only things that can possibly happen - we can't get an equal number of H and T if we flip a coin an odd number of times. And the probabilities of all the things that can happen must add up to 1. So we have two equal probabilities that add to 1, and both probabilities must be 1/2.

Notice that question 2, "what's the probability you get more Tails than Heads" is exactly the question you were answering: it's the same question as: "what's the probability you get at most 2 Heads if you flip a coin 5 times?"

So that's one way to see that the answer is 1/2 here without looking at any cases.

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