Definitions

• equally likely - when each element in S (the sample space) has the same probability of being the outcome of an experiment
• biased - when some elements in S have higher probability of being the outcome of an experiment as compared to other elements in S
• fair - a probability experiment where any outcome in S is equally likely to occur

Requirements for assigning a probability to each outcome in S (the sample space)

• p(s) is a function
  • the domain: all the elements in S
  • the range:
    • p(s) : S -> R⁺ where R⁺ 1
      or it can be written as
    • 0   p(s) 1, s S
• p(s) = 1
• Note, there exists two probability functions p
  1. p(E) - has domain as E, where E S, i.e., E is a set of specific outcomes in S
  2. p(s) - has domain as s, where s S, i.e., s is a specific outcome in the sample space S

Definitions

1. uniform distribution - for sample space S, whose size |S| = n, then s S : p(s) = 1/n
   Note: this is another way of saying that all the outcomes in S are equally likely
2. probability of E - where E S, then p(E) = p(s)
3. random selection - the probability experiment of selecting an element from S where S has a uniform distribution (or all when selecting any particular element from S is equally likely as selecting any other)

Examples

Example #1: Flipping a fair coin

• S = { H, T }, where H represents heads, and T represents tails
• All possible single experiment outcomes: E = { H }or E = { T }
• Assignment of probability:
  • p(H) = 1/2
  • p(T) = 1/2
  • Note: here we are using the function p(s) whose domain is an element s S

Example #2:The probability of rolling an odd number with a biased die

• biased because 3 appears twice as often as each other number
• S = {1, 2, 3, 4, 5, 6}
• All possible single experiment outcomes: E = {1}, E = {2}, E = {3}, E = {4}, E = {5}, or E = {6}
• Assignment of probability (probability distribution for the biased die):
  • p(1) = 1/7
  • p(2) = 1/7
  • p(3) = 2/7
  • p(4) = 1/7
  • p(5) = 1/7
  • p(6) = 1/7
  • Note: here we are using the function p(s) whose domain is an element s S
• Compute probability of E = {1, 3, 5}, or the odds of rolling an odd number with the biased die
  • p(E) = p(s) = p(1) +  p(3) +  p(5)
    Note:
    • the left hand side of '=' uses p(E) whose domain is a set E S
    • the right hand side of '=' uses p(s) whose domain is an element s S
  • p(E) = (1/7 + 2/7 + 1/7) = 4/7
• Note: if the die is fair or unbiased, then all possible outcomes are equally likely, which changes the probability of rolling an odd number:
  • p(s) = 1/6, s S
  • p(E) = p(s) = (p(1) +  p(3) +  p(5)) = (1/6 + 1/6 + 1/6) = 1/2