Probability problem in Hearthstone
Hearthstone is a simple online card game developed by Blizzard Entertainment, a company known to make high quality PC games. Despite its simplicity, there are cases where games can become very complicated, and there are often times when you wonder what the chances of certain events happening are, and I've actually even thought about making an online tool for calculating probability in Hearthstone.
I watch more Hearthstone than I play (mostly due to the existence of secret paladin), and was watching the Curse Trials casted by Kripparian and Frodan when a player was presented three choices out of eight possibilities. Kripparian exclaimed that they were the worst three choices possible, and wondered aloud what the probability of the event was and said it was likely really low. After some thought he came to the conclusion that the chances were about 10%, and that the event wasn't as unlikely as he thought. But is that correct?
Probability can be a tricky area, and I often see Hearthstone streamers incorrectly calculating probabilities, albeit sometimes only barely so. I did not major in maths

^{1}, but I am known to develop new algorithms for probability problems in my spare time, and consider probability to be one of my fortes. The key to probability problems is to convince yourself of the answer, and it easiest to do so when you arrive at the same answer from different paths. It turns out that this problem is actually quite simple. You can imagine that your eight options are in a box and you will be pulling out your three options in the exact same order that the game presented them to you. There are eight items in the box, so the first draw has a chance of 1/8 of being the option the game provided. Now there are 7 options in the box, so the second draw has a chance of 1/7 of being the option the game provided. Similarly, the third draw has a chance of 1/6. So it seems we should multiply these three probabilities, and we should, but we will need to do something else as well. The question wasn't what was the chance of receiving the three worst options in the same order the game presented them, it was just the chance of getting the three worst options, regardless of order. As a result, we need to multiply our probability by how many ways we could have drawn those three options, which happens to be 3!. Multiplying 3! to our probability gives 1.79%, which is not 10%. But how confident are we in that answer? Would we be willing to bet our life on it? I mean the answer feels good, but probability problems can be deceptive. One thing we can do is rephrase the question a little. We wanted three specific options out of eight drawn from a box and don't care about order. Therefore, when we draw the first option we don't care which of the three it is, so it has a chance of 3/8 of being one of our three options. Similarly the second draw has a chance of 2/7, and the third 1/6. Multiplying these probabilities gives us the same 1.79%. That sounds good, but we were really just doing the same method with slightly different wording. In probability the chance of all possible events must sum to 100%, and when I do probability problems I like to enumerate all the events and their probabilities and do a sanity check that they sum to 100%. This is kind of an informal method of proving your calculated probability is correct. To calculate how many ways we can draw three options out of eight, regardless of order, we need to use this thing from high school math class called nCr notation. Basically what nCr notation does is take your eight options, finds all possible ways they can be ordered, then accounts for the fact that you selected three whose order you don't care about, and five options whose order you obviously don't care about. This results in the equation 8!/5!*3!, which is 56. So there are 56 possible ways to draw three options from a box of eight, when you don't care about order. We want to know the probability of one such event out of 56, which is 1/56, which is 1.79%.^{1. I got this spelling from the movie X+Y, I would recommend watching. ↩}