Yes it is possible for an event to have a probability of 1, which means that the event willhappen. It depends on how you frame the question of probability.
For example, if you had a game spinner that had only one color, and the color was red, not matter how many times you spun, the probability that the spinner would land on red would always be 1.
Another example would be: You have a bag of pennies. The probablility of you picking a penny out of the bag is always going to be 1, since the bag has only one choice.