linguistic:translation, stylistic:style_transfer, structural:reformatting
Fallacy: odds are never dynamic

In the beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly.

For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, the probability that the other child is also a girl. Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a probability space illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%).

To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of the children is female, this rules out the boy-boy scenario, leaving only three ways of having the two children: boy-girl, girl-boy, girl-girl. From this, it can be seen only ⅓ of these scenarios would have the other child also be a girl (see Boy or girl paradox for more).

In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information. This technique can be used to provide 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 to 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. However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning.