Tossing coins

The simplest case is a so-called voting ensemble. To illustrate this, think of yourself tossing coins with a biased outcome of 51 per cent for heads and 49% for tails. With only few tosses, you may not clearly see this distribution for heads and tails. However, after some thousands of tosses, there will be a clear majority of heads. With 2000 tosses you should see approximately 1020 heads and 980 tails.

We can then state that the outcome is a clear majority of heads. If you do this ten thousand times, it is easy to see that there is a 97% likelihood of a majority of heads.

Another example would be to collect all polls before an election. Different polls may show different likelihoods for a candidate winning with say a majority of the popular vote. The majority vote would then consist in many polls indicating that this candidate will actually win.

The example here shows how we can implement the coin tossing case, clealry demostrating that after some tosses we see the law of large numbers kicking in.