Probabilities, Cards, and Counting

We started by introducing the concept of probability, or how frequently or likely something happens at.

There is objective and subjective probability, with the former being an actual branch of math, and the latter being a sign of confidence.

The objective probability on an event A can be more precisely defined as the total number of outcomes that satisfy A over the total number of possible outcomes.

There are a few conditions that must be satisfied, but the only important one is that the possible outcomes should occur with equal probabilities.

The 2 rules we use for calculating probabilities are:

Addition principle. The probability of A or B is equal to probability of A + probability of B - Probability of A and B

Mulitiplication principle. For independent events, the probability of both A and B is equal to probability of A times the probability of B.

The probability can be used to calculate the expected values of a lot of things, but sometimes, the number of events that satisfy A can be extremely hard to count.

For example, in Poker, how many distinct three-of-a-kind's are there? 2 pairs? Theoretically one simply needs to go through all possible sets of 5 cards chosen and count, but that takes forever. Therefore, combinatorics is needed.

Firstly, permutations. It refers to the number of distinct ways to pick a group of m elements from a set with n elements, when order in which you picked them in matters. Through algebra derivations, we proved that there are n!/(n-m)! ways to do this.

Secondly, combinations. It refers to the number of distinct ways to pick a group of m elements from a set with n element,s when the order in which you picked them does not matter. Through algebra derivations, we proved that there are n!/(n-m)!m! to do this.

Then, using this, we calculated the probabilities of a hand of 5 cards being a royal flush, straight flush, 4 of kind, full house, etc. all the way to 2 pairs. The exact calculations can be found in https://meteor.geol.iastate.edu/~jdduda/portfolio/492.pdf

Key ideas of this session revolves around the idea of counting so that nothing is overcounted, and nothing is undercounted. Counting in a smart way, for example, by fixing some variables, or doing careful casework, can be very useful both in the real life and solving combinatorics problems.

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