A little background if you are interested:
The binomial distribution is based on yes/no outcomes and does look like a normal curve.
This is based off of many button pushes each of which are called a Bernoulli trial.
The bernoulli distribution (a single trial) might be more what you are envisioning (a peak at 0 and a peak at 1)
However, the expected value for the Bernoulli is still 0.5!
Expected values are a complex bit of stats that isn’t necessarily intuitive. In fact probabilities are rarely intuitive 
Consider that the average family has 1.93 children. This isn’t saying anyone has a kid with 7% missing, but rather describing something about the larger population as a whole. If I look at 100 families I expect to see 193 children. And here if I look at 100 50/50 button pushers I expect 2.5 billion in total funds across all of them. So 25 million each.
This group, where the average person has 25 million, is also an excellent example of why we need to understand what a statistic is telling us. For example, someone (guess which party) could intentionally mislead you to beleive that no one is poor because the average person has 25 million.
