Probability 확률

목차

Probability Theory 확률

Probability: science of uncertainty

 

1. Classical Probability Theory

[Equal-Likelihood Model] ~ selected “at random”

      N possible outcomes, event that can occur in f ways

P(A) = f/N

• Experiment: process / situation in which different things can happen
• Outcome: one of the possible things that can happen
• Outcome space (sample space): the set of all possible outcomes of an experiment
  • S = {H, T}
    • Outcome in sample space are always mutually exclusive & exhaustive (=covers all possibilities)

A = (roll even number) = {2, 4, 6} A : event (roll even number) : event defined {2,4,6} : outcomes; outcome space

2. Frequentist (empirical) Probability

• Do the experiment → get probability
• Not exactly 50-50 chances (it’s biased)
  • Ex) certain baseball team winning

3. Subjective Probability

• Probability as degree of belief that an event will occur or that a proposition is true
  • Ex) the probability that this person’s lying is 97%!

* Special Type of Event

• ∅ = { }
  • Null set; impossible set

Compound Event

1. Intersection: A and B = A ∩ B
2. Union: A or B = A ∪ B
3. Set Complement: Not (A) = Ac = A'

• Other Wording
  • at least X:    ≥ X
  • at most X:    ≤ X
  • between X and Y:    X ≤ ** ≤ Y

Probability (Equally Likely)

If outcome space S consists of a finite # of equally likely individual outcomes, and A represents any event, then

P (A) = # of outcomes in A / # of outcomes in S = #(A) / # (S)

• Ways of finding complete outcome space (# of outcomes)
  • 1) graphical devices 
    • ex) outcome trees
  • 2) combinatoric fromulas: figure out many outcomes in an outcome space
    • in a sequential or multi-stage experiment, there are n possibilities: #(S) = (n1)(n2) ...
      • ex) with replacement: #(S) = 7 * 7 * 7
      • ex) without replacement: #(S) = 7 * 6 * 5
    • 3) ordered - permutation of n objects = n!
      • incomplete permuation (ex. randomly select 3 books to be 1st, 2nd, 3rd place from 7 books)
        • nPr = n! / (n-r)! → (short cut) = (n)(n-1) ...until the number of r
          • ex) 7P3 = 7 * 6 * 7
    • 4) order is not important - combination (like subcommittee problem)
      • (ex. select 3 people from 10)
        • nCk = n! / k! (n-k)!
          • ex) 10C3 = 10! / 3! (10-3)!

Probability (Unequally Likely)

P(A) = k Σ (i =1) P(ai)

* probability of all outcomes sum to 1

 

	Male	Female
Psychology Major	(joint probability)	(joint probability)	<marginal probability>
Math Major	(joint probability)	(joint probability)	<marginal probability>
English Major	(joint probability)	(joint probability)	<marginal probability>
	<marginal probability>	<marginal probability>

• P (A or B) = P (A ∪ B) = P (A) + P (B) - P (A ∩ B)
  • if A and B are mutually exclusive (A ∩ B = ∅), then  P (A ∪ B) = P (A) + P (B)
• P (A') = 1 - P (A)

Conditional Probability

probability that event B occurs given that event A occurs

= P (A | B) = P (A ∩ B) / P (B)

So, P (A ∩ B) = P (A | B) * P (B)

 

• Independence
  • two events are independent if,
    • P (A | B) = P (A) OR P (B | A) = P (B)
    • P (A ∩ B) = P (A) * P (B) 
  • occurring A or not makes no difference as to how probable B is
    • ex) drawing a card from a deck of card with replacement would not affect the 2nd drawing
  • if A & B are mutually exclusive, then A and B are NOT independent