I assume there are various ways of explaining it, but what's the most intuitive (yes I understand that is a relative concept)
Edit due to request:
Obvious example: There is a 1/6 chance of rolling a 6 on a 6-sided die. Expected number of rolls is 6/1.
Another example: You have 4/6 chance of rolling 1-4 on a 6 sided die. Expected number of rolls is 6/4.
Another example: There is a 2/52 chance of picking a red queen from a deck of cards. Expected number of draws (with replacement) is 52/2.
edit: Please replace "odds" in my title to "probability". As pointed out in a reply, I did not use the right word.