(skewness of) binarized multinomial / normal distribution

The binomial distribution has the interesting property that it is skewed for probabilities unequal. For me as a decision scientists this means that if $$100$$ individuals repeat the same bernoulli experiment $$10$$ times, then the majority of individuals will (on average) see the event less often than expected if $$p < .5$$ and more often than expected if $$p > .5$$.

This has fundamental implications for how we (as humans, organizations, etc.) behave in environments with rare events. Namely we can expect that the majority of individuals will be thinking that the rare event is less likely than expected (provided we are in a world governed by the binomial distribution). Now the question is can this reasoning be extended beyond the binomial-distribution-world? For instance, if one was to binarize a multinomial distribution into the probability of one event versus the rest, would the resulting distribution also be skewed (when viewed as a function of the events probability)? Or, if we sample from a normal distribution, and we ask whether individuals see events more extreme than 3SD (on average) more often than their to their expectation (i.e., their relativ mass in the distribution) or less often.

• The value of the parameter $p$ does not change the expected number of events. The number of events is always expected to be $np$ for $0<p<1$. Having $p<0.5$ does not mean individuals will see the event "less often than expected." – StatsStudent Feb 23 at 18:46

This is not true. Let $$n=100$$ and $$p=.3$$. Then $$X$$ (the number of individuals seeing the event) is $$\mathcal{Bin}(100,.3)$$ with expectation $$n\cdot p=30$$ so the expected number of individuals seeing the event is correct. The expectation of a binomial variable in general is $$n p$$ so what you say will never be true. (And this really has little to do with binomial distributions, it has to do with the law of large numbers.)
So you are misinterpreting skewness. What does the skewness of the binomial distribution mean? Skewness is simply a standardized third moment. For the binomial it is $$\frac{1-2p}{\sqrt{np(1-p)}}$$. Below a histogram of the binomial distribution for $$n=100,p=.3)$$:
The expected value bar (30) in red. Positive (small) skewness. Compare the bars on each side of 30: The bars above is lower that the bars below. Skewness means that when the observed value is larger than expected, it tends to be further above than when it is below. This is then matched by lower probabilities. But in the long run it averages out, by the law of large numbers. Another way of looking at this is by partitioning the variance: $$\sigma^2 = \sum_{k=0}^{100} p_k (k-30)^2 = 21$$ Let us take the sums above and below 30 separately: \begin{align} \sigma^2_{lower} &= \sum_{k=0}^{30} p_k (k-30)^2 = 10.25587 \\ \sigma^2_{upper} &= \sum_{k=30}^{100} p_k (k-30)^2 = 10.74413 \end{align} confirming the slightly larger spread on the upside. That is the meaning of skewness.