probability of at least (k) URs in (n) tries

In the beginning of a scenario, one might calculate the probability of a certain event. The fact is, as soon as one gains more information about that situation, they may need to re-calculate the probability.

Say we are told that a woman has two children. If we ask whether either of them is a girl, and are told yes, what is the probability that the other child is also a girl? Considering this new child independently, one might expect the probability that the other child is female is ½ (50%). But by building a probability space (illustrating all possible outcomes), we see that the probability is actually only ⅓ (33%). This is because the possibility space illustrates 4 ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But we were given more information. Once we are told that one of the children is a female, we use this new information to eliminate the boy-boy scenario. Thus the probability space reveals that there are still 3 ways to have two children where one is a female: boy-girl, girl-boy, girl-girl. Only ⅓ of these scenarios would have the other child also be a girl. Using a probability space, we are less likely to miss one of the possible scenarios, or to neglect the importance of new information. For further information, see Boy or girl paradox.

This technique provides insights in other situations such as the Monty Hall problem, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prizes behind the others. Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left (one with the car, the other with another goat), the player must decide to either keep their decision, or switch and select the other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference. But probability spaces reveal that the contestant has received new information, and can increase their chances of winning by changing to the other door.