Probability is the measure of how likely an event is to occur, typically using decimal values with 0 signifying an outcome is impossible, 1 signifying a guaranteed outcome, and every decimal value in between representing the chance of that event happening.

The most simple way to calculate probability is using a basic fraction. For instance lets ask what the probability is that we roll a D6 (six sided die). There are six possible outcomes and only one will be the result of each throw. \[\mathbb{P}(Roll\ a\ 6) = \frac{1}{6}\]

Set theory involves grouping outcomes into “sets.” For instance if we poll a university on what majors students are studying (Engineering, Science, Math) we can group students’ answers into sets. Lets say for example ten students (numbered 1-10) are asked the aforementioned question, 5 are engineers, 3 are scientists, and two are mathematicians They can be grouped as engineers : {1,2,3,4,5}, scientists: {6,7,8}, mathematicians: {9, 10}. Thus organizing the various groups into sets. lets say students 4 and 5 are also scientists, 5 and 8 are mathematicians, and 9 is an engineer. To visualize this, we place the sets into Venn diagrams.

engineers = c(1,2,3,4,5,9)
science = c(4,5,6,7,8)
math = c(5,8,9,10)
majors = list(engineers= engineers,
              scientists=  science,mathematicians= math)
ggvenn(majors, show_elements = T)

Conditional Probability is the likelihood of an event happening given that another event has already occurred. Sometimes an event is more likely to occur if something else already has. For instance the probability that you cough on a normal day is low, but if you are sick, it is more likely for you to cough. Therefore the probability that you cough given that you are sick is higher than the base probability of coughing. Conditional probability is calculated and notated the following way: \[\mathbb{P}(A \mid B) \ = \ \frac{\mathbb{P}(A\ \cap \ B)}{\mathbb{P}(B)}\]

In Venn diagrams the intersection of two circles signify that both events are true. This intersection gives us the numerator of the previous slides equation. All numbers in the circle signifies the event occurring, giving us the denominator to the previous equation. Using both of these we can find probabilities and conditional probabilities from a Venn diagram.

If we use the data from the last slide we can find conditional probabilities representative of the students and their majors. \[\mathbb{P}(Engineer\ \mid \ Scientist) = \frac{\mathbb{P}(Engineer\ \cap Scientist)}{\mathbb{P}(Scientist)}\] We can use the Venn diagram to solve this. There are two people in both the engineers circle and scientists circle, and 4 in the whole scientists circle. Using this information we form the equation. \[\mathbb{P}(Engineer\ \mid \ Scientist) = \frac{2}{4}\]

Venn diagrams aren’t the only way to show conditional probabilities, probability trees share conditional probabilities without having to calculate it. With the branches representing conditional probabilities and the nodes representing the outcomes. Lets make a tree that shows two coin flips where heads has a 55% chance of happening.

Where Venn diagrams give us base and joint probabilities to help find conditional probabilities, trees give us base and conditional probabilities to help find joint probabilities. Manipulating the equation earlier with algebra to accommodate for the change of information we are left with this: \[\mathbb{P}(H\ \cap \ T) = \mathbb{P}(H) * \mathbb{P}(T\ \mid \ H)\] using the numbers from the tree we get: \[\mathbb{P}(H\ \cap \ T) = (.55)*(.45) = .2475\]

I hope you enjoyed my crash course in probabilities!