When intuition and math probably look wrong (2010)
It always helps me to picture a probability space (as in, an actual physical space, where size correlates to probability).<p>Assuming that children are always a boy or a girl, and are equally likely to be a boy or a girl, the &#x27;complete space&#x27; of families with two children would be distributed as
ABBC
where A means two boys, B means a boy and a girl, and C means two girls. By excluding C from the space, ABB remains. So indeed, the probability would be 1&#x2F;3 that both children would be boys.<p>But now I have made another assumption, namely that the family is picked uniformly random from all families with two children with at least one boy.<p>As you can see, there are quite a lot of assumptions. One convention seems to be that when you can&#x27;t tell for sure how something is distributed, it is uniformly distributed.<p>For example, when a person blindly grabs a ball from a box with one red and one blue ball (and no other balls), the probability is not always 50% that he grabs the red ball. The red ball might be a bowling ball and the blue one might be the size of a marble.<p>While in this example it is reasonable to assume a uniform distribution, you can point out similar, but more subtle assumptions in most questions about probability.<p>A more complicated example:
Weatherman A predicts the weather right 70% of the time.
Weatherman B predicts the weather right 60% of the time.
Weatherman A predicts rain tomorrow, weatherman B predicts dry weather. What is the probability that it will rain tomorrow?<p>The &#x27;right&#x27; answer is 14&#x2F;23. In order to arrive at this answer, you need to assume that the predictions are statistically independent (which is an unrealistic assumption). Indeed, it is easy to sketch a situation in which the probability is different: it is always dry, weatherman A predicts rain 30% of the time, and weatherman B predicts rain 40% of the time. This is consistent with the question, and the probability that there will be rain tomorrow is obviously 0%.<p>This lack of rigor always bothers me.