039 Probability

039 Probability

1. Sample Space - the set of all possible outcomes of a random process/experiment

2. Event - subset of sample space

3. Theorem: P(E) = the number of outcomes of E / the number of outcomes in S

   • Example: A fair coin is tossed 7 times
     • What is the probabilty that we get exactly 5 heads?
       • P(5 heads) = (7C5) / 2^7 = 0.16
     • What is the probabilty that we get at least 5 heads?
       • P(>=5 heads) = (# with 5 H) + (# with 6 H) + (# with 7 H)
         • = 7C5 + 7C6 + 7C7 = 21 +7 + 1 = 29
           • P(>=5 heads) = 29/128 = 0.23
   • Example: Suppose there are 23 people in the room. What is the possibility that there are at least two people who share the same birthday?
     • Sample space = 365^23
       • Let E be the event that at least two people share the same birthday
         • Let Ec be the event that all 23 people have different birthdays
           • P(E) = |S| - |Ec|/|S| = 1 - 0.493 = 0.507
           • Ec = |S| - |E|
         • E = 365 * 364 * 363 * 362 * ... * 343

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