Probability

• A probability is a number that is assigned to an event, and it describes how likely it is that the event occurs.
• A probability is a number between 0 and 1
• An event with a probability of 0 will never occur
• An event with a probability of 1 will always occur.
• In Theoretical probability we compute the probability of an event based on the sample space or the list of possible outcomes.
• For a given experiment with sample space S, the probability of event E occurring P(E) = n(E)/n(S)
• Examples:

  Toss a 12 sided die
    What is the sample space?   S= {1,2,3,4,5,6,7,8,9,.10,11,12} n(S) = 12
  
  Let E be the event that a 3 is rolled
    What is E?  E={3}, n(E) = 1
    P(E) = n(E)/n(S) = 1/12 = 0.083   
  
  Let E be the event that an even number is rolled
    What is E?  E={2,4,6,8,10,12}, n(E) = 6
    P(E) = 6/12 = .5
  
  Let E be the event that a number less than 6 is rolled
    What is E? E={1,2,3,4,5}, n(E) = 5
    P(E) = 5/12 = .41
  
  Let E be the event that a number larger than 0 is rolled
    What is E? E={1,2,3,4,5,6,7,8,9,10,11,12}, n(E) = 12
    P(E) = 12/12 = 1
  
  Let E be the event that a number divisible by 13 is rolled
    What is E?   E = {}, n(E) = 0
    P(E) = 0/12 = 0
      

• odd problems 42-48 page 585

  S = red x 3, , brown x 3, blue x 2, yellow x 2.
  n(S) = 10
  
  P(yellow)  = 2/10 = .2
  P(brown) =  3/10 = .3
  P(yellow or brown) = 5/10 = .5
  P(yellow and brown) = 0/10 =  0
      

• Number 28

  Why do we have 8 choices?  (FCP)
  n(S) = 8
  
  E = exactly one male child.
  n(E) = 3
  P(E) = 3/8 = 37.5%
      

• Empirical probability is based on the conducting the experiment over and over again.
• We need to know the sample space

•     
               number of times that event E occurred
      P(E)  = --------------------------------------
               total number of observed occurrences
      

• For the data for problem 56,

  Let E be the event that a person bikes
  P(E) = 240/2000 = 12%
  
  Let E be the event a person bikes or uses exercise machines.
  P(E) = (240+220) / 2000  = 23%
      

Homework