
Probability is usually taught using dice and coins. In this post, we are going to do things a bit differently. We are going to use examples from Harry Potter instead. Today we will focus on dependent random events that took place during the Triwizard Tournament in Harry Potter and the Goblet of fire.
Dependent events rely on events that have come before them. For instance, many actions in card games can be considered as dependent events because the probability of drawing cards from the deck depends on the cards that have already been played. If I were to count cards in the game of Black Jack I would need to remember all the cards that have been played before the deck is reshuffled in order to analyze my probability of winning a hand.
Think of Conditional Probability as a way to analyze dependent events. We denote the conditional probability as ℙ(A|B) which means the probability of event A occurring given event B has already happened.
Prerequisites & Notation
- Ω: sample space; considered a set.
- ω: sample outcome.
- A: event; a subset of Ω.
- |A|: the number of elements in set A.
- a ∈ S: the value a is "a member of" or simply "in" set S.
Recall, the Triwizard Tournament example from the previous post where our sample space was Ω = {SSS, CWG, CF, HH}. In order to illustrate the effects of dependent events let’s change the sample space to the following: Ω = { SSS, SSS, CWG, CWG, CWG, CF, HH} where
- SSS = Swedish Short-Snout
- CWG = Common Welsh Green
- CF = Chinese Fireball
- HH = Hungarian Horntail
The chances of choosing a Swedish Short-Snout are 2 in 7. If Fleur Delacour chose Swedish Short-Snout, what are the chances of then choosing a Swedish Short-Snout? Well, the chances have changed since we removed a dragon from the bag because there’s now one less.
The idea here is that if we replaced the dragons each time a wizard chose one, then the Probability would not change and the events are considered independent. However, in this example the dragons are not being replaced, thus the events are considered dependent.
Definition: If ℙ(A) > 0 then the conditional probability of B given A is

Let’s discuss this notation a bit more.
In our example let’s say event A is "choose a Swedish Short-Snout first" with a probability of 2/7 and event B is "choose a Swedish Short-Snout second".
Now here is the tricky part, for the event B we have four choices:
- If we chose a Swedish Short-Snout first the probability is now 1/6.
- If we chose a Common Welsh Green the probability is now 3/6.
- If we chose a Chinese Fireball the probability is now 1/6.
- If we chose a Hungarian Horntail the probability is now 1/6.
We have to stipulate which event happened first. To do that we use the mathematical symbol "|" to denote "given". Thus, ℙ(A|B) means "Event A given event B" and is also referred to as the conditional probability of A given B. Let’s do an example using our fancy new notation. What are the chances of choosing a Common Welsh Green dragon first (Event A), and then choosing a Common Welsh Green second (Event B)? So we can start with the conditional probability formula and rearrange it a bit:

We know the chances of choosing a Common Welsh Green is 3/7 (ℙ(A)), but after we remove that dragon from the bag, the second dragon chosen from the bag is less likely to be a Common Welsh Green: ℙ(B|A) = 2/6.
So we plug in the numbers:

This may seem simple, but sometimes it can be tricky. For instance, it is usually not the case that ℙ(A|B) = ℙ(B|A). In fact, they can be extremely different! Let’s think about this when answering the question: What is the probability of choosing a Swedish Short-Snout given that we chose a Chinese Fireball first, ℙ(A|B)?
Secondly, is this probability the same as choosing a Chinese Fireball given a Swedish Short-Snout was chosen first (ℙ(B|A))?
Well, the answer to the first question is 2/6, since there are two Swedish Short-Snout’s in the bag once we removed the Chinese Fireball. Now, if we chose the Swedish Short-Snout first and then chose the Chinese Fireball, the probability of this event would be 3/6. See! Not the same! ℙ(A|B) is not equal to ℙ(B|A)!
In the next Triwizard Tournament, you will know exactly what your probabilities of choosing each dragon will be. Hopefully, this new skill will help you make it past task one.