Mathematical Interlude — Probability
Introduction
Well I don’t mean to interrupt the chapter on stars but I really feel like covering a few more important topics in Maths might actually be really helpful.
Today we’ll be doing probability. I personally hate the topic a lot, mainly because it seems kind of counter-intuitive and confusing but in reality the principles of Probability are quite fundamental.
Now by the end of this post, I guarantee that you will not be able to win a jackpot at a Las Vegas casino but you will end up walking away with some knowledge of basic probability.
“Probability theory is nothing but common sense reduced to simple calculation.”
- Pierre-Simon Laplace
I agree with this quote to a certain extent but once we get into the nitty, gritty (which we won’t today), probability starts to become a bit harder. Anyway let’s get started.
Probability is the study of chance, rather it looks at how likely an outcome is based on a given scenario. In fact a line can be drawn up to quantify the possible “outcomes” in a probability or rather what the outcomes mean. The line is from 0–1 and looks like this:
In fact based on this line, you can see that the phrase “100% guaranteed” doesn’t start to make sense. As my seventh grade Maths teacher would say, “If something is guaranteed, it’s already a 100%. Adding a 100% in front of it makes the phrase redundant.”
Hold on. How does having a probability of 1 mean something is guaranteed, wouldn’t having a probability of 100 mean something is guaranteed?
The answer to that is actually it depends. If you’re looking at a scale from 0–100 then you’re right. However convention states that we look at probabilities as a fraction. In fact we can represent the probability of any event happening as a fraction like this:
Let’s suppose we’re rolling a six sided die and we want to find the probability of getting a 5. Now you might be inclined to say that the probability is fifty-fifty, you either a roll a 5 or you don’t. That’s wrong, let’s look at the formula. How many desired outcomes do we have? 1.
The only desired outcome is to roll a 5 and so you only have 1 desired outcome. How many possible outcomes do you have? You have 6. You can roll any integer from 1–6 and so you have 6 possible outcomes. Therefore your probability of rolling a 5 would be 1/6. This means that rolling a 5 is unlikely.
Now what if we wanted to find the probability of rolling an even number? Let’s look at how many desired outcomes we have. The even numbers between 1 and 6 are 2, 4 and 6. Therefore we have 3 desirable outcomes and 6 possible outcomes. Therefore the probability of rolling an even number is 3/6 or 1/2 which just happens to be fifty-fifty.
Let’s do one more. We want to find the probability of rolling an integer. Now an integer is any whole number that can be positive, negative or 0. That means any number from 1–6 is going to be an integer. Therefore we have 6 desirable outcomes and 6 possible outcomes. Therefore the probability is 6/6 or 1. This means that rolling an integer is guaranteed.
The three rules of probability
- The probability of any event happening must be between 0 and 1, the probability of something happening cannot be negative nor can it be greater than 1. (What does it mean to have a -0.73 chance of rolling a 5?!)
- The sum of all probabilities must be 1. (If I add up the probabilities of getting a 1, 2, 3, 4, 5 and 6 on a six sided die, I’ll end up getting 1.)
- The third rule is pretty similar to the second, if I add up the probability of one event happening and the probability of it not happening, it should be 1. (The probability of rolling a 6 is 1/6 and not rolling a 6 is 5/6. 1/6+5/6 = 1)
Sample Spaces
So far we’ve dealt with the easy probability. We’ve only had to deal with 1 die. What if we have a pair now? Let’s suppose I’m playing monopoly and I want to roll a double (both my die roll the same number). How can I find the probability of rolling a double. This seems kinda confusing and this is where a sample space can come into help. We use sample spaces when we’re dealing with two events (Each die can be considered a separate event) but they can be extended to more.
So what is a sample space? A sample space is just a list of all the possible outcomes, we typically represent it on this coordinate axis like graph. On the x axis, write out all the possible outcomes from die 1 and the same on the y axis for die 2. Your sample space should look something like this:
We can fill this up to show:
(1, 1) represents getting a 1 on both die. Therefore the desirable outcomes are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6). So there are 6 desirable outcomes and 36 possible outcomes. So the probability of rolling a double would be 6/36 or 1/6. This means that rolling a double is unlikely.
A sample set for a deck of cards looks like this:
Tree Diagrams
Tree diagrams are generally used when you have more than 2 events but we could also use it for the above example. Essentially we draw these long lines called a tree, each action or outcome is represented as a branch. Essentially it looks like this:
Now we can see that to get a double. We have to either roll a 1 and a 1, a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5 or a 6 and a 6.
Notice my use of the words and and or carefully. Therefore the probability of rolling our first 1 would be 1/6. The probability of rolling our second 1 would also be 1/6. Now if I use the word and, I have to multiply my two probabilities. So 1/6 * 1/6 becomes 1/36. I repeat the same for the remaining 5 and I get.
- Probability of rolling a 1 and a 1: 1/36
- Probability of rolling a 2 and a 2: 1/36
- Probability of rolling a 3 and a 3: 1/36
- Probability of rolling a 4 and a 4: 1/36
- Probability of rolling a 5 and a 5: 1/36
- Probability of rolling a 6 and a 6: 1/36
Now I would be satisfied if I get either of these. So if I get the first outcome or the second or the third and so on, I would be happy. If I have the word or, I just add up my probabilities. So now I just add up my six probabilities to get 6/36 which is 1/6.
Probability with and without replacement
Let’s say I have a bag which contains 10 marbles. 7 are black and 3 are red. I want to find the probability of getting a black and a red marble in that specific order. So I want to get a black marble first.
There are 7 black marbles and so there are 7 desirable outcomes. There are 10 marbles and so 10 possible outcomes. Therefore the probability of getting a black marble is 7/10 and a red marble is 3/10.
Since I want a black and a red marble together, I have to multiply the two probabilities to get 21/100, this means I have a 21% chance that if I pick two marbles randomly, I will get a black and a red marble in that order.
Now what if after picking a marble, I don’t put it back in the bag? This is known as probability without replacement. Let’s use the same scenario where I want a black and a red marble in that order. My probability of picking a black marble is 7/10. However now I don’t replace this marble in the bag, I remove it permanently.
Since I want a red marble and there are 3 of them, there are 3 desirable outcomes. However now there are only 9 possible outcomes as I removed one marble. Therefore the probability of picking a red marble is 3/9 or 1/3. Since I want both a black and a red marble, I have to multiply the two probabilities to get 7/30.
Law of Large numbers
All of what we’ve looked at so far is just theory. If I had an actual bag of black and red marbles and if I keep picking marbles at random with replacement, it’s possible that I could get a red marble every time. It’s unlikely but it’s still possible. However there’s a way to tie theory and real life together and that is via the law of large numbers.
The law basically states that if I do an experiment many times, the results become closer to what is actually expected. This means that if I do the marble experiment 10 times, 100 times, 1000 times, I might still keep getting red marbles but the more times I do it, the more the probability of picking a marble gets closer to theory.
I hope you enjoyed this brief intro into probability. Probability can get more complex if you combine it with Set theory and look at things like Bayes’ rule but at the same time that’s when probability starts to get fun.
Thanks for reading!
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