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Had an interesting example the other day from basic renewal theory that can easily be explained without doing any math:
Suppose you're rolling a fair 6-sided die and you're interested in average number a trials to get a certain sequence, lets make it only 2 rolls long for simplicity, like "34". By grinding the math (involving probability generating functions) it can be shown that on average it takes 36 trials for "3 4" to appear, which totally makes sense ( since P(any particular #) = 1/6 => 6 trials for 1 number, 6^2 for two). But if you're waiting for "33" for example, surprisingly it takes 42 on average. (This was kind of funny cause prof took a poll before giving an answer and majority of class voted wrong - for needing less trials than 36). It has a really simple logical explanation though - if anybody wants to give it try.
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