Suppose that we roll a fair die until a 6 comes up or we have rolled it 10 times. What is the expected number of times we roll the die?
The answer to this is about 5.03.
However, if you remove the restriction of rolling it 10 times, the answer is 6.
I'm having a difficult time understanding conceptually why it would be different for infinite amount of rolls vs a max of 10 rolls. Both of the expected values are under 10, so why does it change so much?