I recently had a brief exchange with someone on Twitter, who mocked someone coming up with a probability of each team getting through their first group of the World Cup that added up to 200%. At first look, the probability calculations made sense to me since two out of four teams qualify in each group, the one that comes first and the one that comes second (which are mutually exclusive). But he insisted that by definition the probabilities should make 100% (or 1, if you like) and should be normalised. I know what he means about adding to 100%, but I still think one of us are missing something - I’m guessing it’s me.
So, by way of analogy, imagine:
There are going to be two prize draws for first and second place. If you get first place, you can’t draw again for second place - it’s exclusive, you come first or runner up but not both.
If four people take one of 4 cards from a bag, the probability of selecting the one winning card is 25%.
Then the remaining people get to see who comes second by taking one of 3 cards from a second bag, with one winner again (probability approx. 33.3%)
If we say the first and second winner are both ‘the winners’ then the probability of any one person being one of ‘the winners’ is 0.5 (calculated from 0.25 + (0.75 * 0.33) which is the probability of either winning the first draw or losing the first draw and winning the second). This makes sense, because it’s two winners out of four and the outcome is determined via random chance.
When you add the four people’s probabilities you get 200% but that seems to be because the term ‘the winners’ is descriptive of two mutually exclusive events (coming first or coming second). The result still seems to make perfect sense, though. I don’t see how you could normalise those probabilities to 100% and get anything remotely sensible. After all, they all have the same chance of being one of ‘the winners’ and it isn’t 25%.
So, for the World Cup example, it seems like the same situation. Coming 1st or 2nd is exclusive, and ‘qualifying’ describes the teams that either of those seperate events happens to. If you imagine that getting 1st or 2nd was random chance (instead of skill based, etc) like bag-of-cards example, you’d expect a 50% chance of each of the four teams getting to the next round, and that would still add up to 200%.
Where adding up to 100% (or 1) comes in seems to be that each team has a chance, p, of being one of the two qualifiers and 1-p chance of failing to qualify.
Is my analysis of the situation correct, or am I missing something?
