📗 -> 10/1/24: Intro to Sets
🎤 Vocab
S is once again our sample space
The following are defined below:
Basic Rule of Counting - If trials are independent then can be multiplied. Formula below.
Permutations, Combinations, Naive Definition of Probability,
❗ Unit and Larger Context
Important Set Equations:
✒️ -> Scratch Notes
Ex) How many binary passwords of length 6 contain either 3 0s at the start, or 2 1s at the end
A = 3 0s at start
B = 2 1s at end
A = 2^3
B = 2^4
Basic Rule of Counting - n stages, outcomes of stage i is equal to
Permutation - Given n items wish to select/pick r of them where order matters. The total # of posible outcomes is equal to:
Combinations - n items in total choose r of them order does not matter. How many outcomes are possible?
- When order doesn’t matter, you have to remove combinations that are the same. This takes the place of r, reducing the possible permutations.
Ex) Meeting 230 people, 3 conference rooms
Room A fits 30, B fits 50, C fits 150
Choose one, subtract the people, then choose the next one. Order you do it in doesn’t really matter, as the multiplications will cancel.
Continuing on, how many combos of Joey and Schmoey don’t put them in the same room?
Drawing it out, and thinking about how two people are guaranteed to be together helps.
Probability
Naive Definition of Probability - Let S be the sample space of an experiment, and E be an event (
However:
- Not all outcomes E are equally likely.
- Sample spaces can be infinite
This leads to our constraints for using the naive definition of probability: - Each outcome must be equally likely
- Sample space must be finite
Ex) Prob single dice roll P(x=4)
Frequentist / Experimental:
- An alien would observe a large time scale of trials, and record the occurrences vs not occurrences
- Eventually, the observed probability will approach true frequency
P(A) = long run fraction of: A occurring / trials
🧪-> Example
- List examples of where entry contents can fit in a larger context
🔗 -> Related Word
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