An important event is taking place today: The US elections. Americans will vote and the world will soon find out who is destined to be the next American president. The first woman ever to do so, Hillary Clinton; or alternatively Mr. Trump.
An American election is a complicated process. It isn't just a matter of counting up the total number of votes over the country. Aggregation of districts and states plays a role together with other subtleties. Have you discussed the election with your little ones at home? If you are interested in engaging your children's mathematical thought, you can perhaps build on the fishing example that we recently presented.
As a simplification, ignore states, districts and other candidates (other than Hillary and Trump). Just think of a single lake with 1000 fish (voters). Some of them will vote for Hillary, others will opt Trump. So say that there are H Hillary fish (voters) and T Trump fish (voters). Then clearly,
H + T = 1000.
In contrast to the previous fishing example where only some of the fish were fished out, in this situation assume that all 1000 fish are being pulled out (all votes are cast). As they are fished, the votes are sorted according to Hillary type and Trump type. At the end, if it is evident that H > T, America gets the first female president (it is about time)! Otherwise, Trump.
In our lake, let's assume H > T. For example, assume that H = 550 and T = 450. In this case, Hillary will win the election. Now here is an interesting question:
What is the chance that throughout the fishing processes, the number of Hillary fish is greater than the number of Trump fish?
The point is, that even if Hillary wins it is possible that she will not lead the whole way. For example a possible fishing (voting) scenario is:
First 300 votes: All Trump.
Next 550 votes: All Hillary.
Final 150 votes: All Trump.
In such a scenario, Trump appears to lead during the first 599 votes! That can be horrifically exciting!
There are obviously many other sequences in which Hillary and Trump fish can be pulled out. A huge number actually! The question is then, out of all of the possible voting sequences, what is the chance that Hillary leads throughout?
It turns out that this is a famous problem in classic probability and combinatorics. It is called the ballot problem. It is an example of a situation where rather complicated mathematics yields a very simple solution. The result is as follows:
The chance that Hillary leads throughout
the fishing process is: (H-T)/(H+T).
In our numerical example this translates to (550-450)/1000 = 10%. That is, there is a 10% chance that Trump never has the lead.
Conversely, there is a 90% chance that Trump will lead the counting process for a bit, even though he will eventually loose. Quite exciting!
Do you find this interesting? If you discussed this with your kids, let us know. We would love to hear how receptive your children were to this concept.
Let's hope for some good fish.