Precalculus (7)

Bayes Theorem process

• Determine all branches and possibilities for each branch

• Find what you are solving for, could be two P() or just one, make sure to underline key terms in order to decide if it is given, and if so of what

• Form the formula for that and then determine probabilities using multiplication/addition rules

What are the two main types of problems this unit and why are they solved that way?

• Forumla-based

  • Simply forming integer/nCr/etc fractions, adding decimals

• Free-thinking/straightforward

  • Quite a few can be solved with this, some can only be solved like this

  • Think logically, these rely on you not overcomplicating things and just thinking it out

    • Like the birthday problem, just thinking what’s the possibility person B can have a B-day A didn’t have and so forth, and since choosing a day is x/365, each person is just a total permutation of 365C#/365^#

Why are some probabilities multiplied and others added?

• Multiply when you want to find the probability of one thing AND choosing another thing alongside it (the probability of two independent events happening together)

  • Prob. that I get service I and it fails is the same as chance I choose service I and then choose fail!

• Add when the two events cannot happen together (mutually exclusive)

  • When finding total probability of failing, I’ll need to find every possibility of every branch sequence that ends in failure. These P(A and B)’s are each part of the whole which cannot happen simultaneously so when I add them I get the probability that A or B fail AKA the general probability of failure

What is probability?

Favorable/Total possibilities

• Applying this to situations

  • Permutations/combinations would be specific arrangements/total arrangements, etc.

How do you find P(A) or P(B) and explain why it works

P(A) + P(B) - P(A and B)

this is essentially the formula form of a venn diagram, so we subtract the elements that are counted twice by subtracting the intersection, same thing as doing (5 - 1) + 1 + (7-1) = 5 + 7 -1 (when simplified)

how do you find P(A | B) and why does it work

P(A and B)/P(B)

again, this is just a venn diagram in numbers form. In the diagram, given basically means how many A are also in B and what is the probability of find those, which is just the intersection over the total in B

What does “at least” signify?

Subtraction!

At least 1 means you’ll just want to find 100% minus the probability you choose none

Essentially, subtract the complement from 1

What does “at most” signify?

Same thing as least, subtract the complement!

Though sometimes it can be more tricky, for example:

At most 4 juniors, I must subtract from one the probability I get 5 or (more) juniors.

What is sample space and some tips for it?

underline key words that you are solving for (remember given incident)

When you set up a table of all possible values, each side of values on the table can be 4 for 4 sided die, 2 for two sided coin, etc.

Don’t miscount, go row by row or column by column as opposed to counting diagonally

Independent events vs arrangements

In problems, since indpendent events are, well, independent, the probability each specific event happens must first be calculated and then multiplied for the total probability (if needed)

Arrangements are just the number of needed arrangements over the total arrangements

A DIE ROLL WOULD BE AN EXAMPLE OF INDEPENDENT EVENTS, rolling a 1,2, and 3 in that order would be the same as 1/6 for the 1, then 1/6 again for the 2, etc.

(revise this wording)