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Lecture 1

Lecture 1. Sec 1.1-1.4. Outline. Define terms with an example (outcome, sample space, event, an event space) Axioms of probability. Ficticious Experiment. Experiment : flip a coin three times. Observe the sequence of heads and tails .

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Lecture 1

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  1. Lecture 1 Sec 1.1-1.4

  2. Outline • Define terms with an example (outcome, sample space, event, an event space) • Axioms of probability

  3. Ficticious Experiment • Experiment: flip a coin three times. Observe the sequence of heads and tails. • Question: Assuming that the coin is fair, what is the maximum number of heads and tails?

  4. Ficticious Experiment • Experiment: flip a coin three times. Observe the sequence of heads and tails. • Results: • hhh • hht • hth • htt • thh • tht • tth • ttt outcome sample space Definition of an outcome: any possible observation of the experiment Sample space: a set of all possible outcomes.

  5. Another Example

  6. Ficticious Experiment • Procedure: flip a coin three times. Observe the sequence of heads and tails. • Results: • hhh • hht • hth • htt • thh • tht • tth • ttt An event is a set of outcomes with the first flip being a head.

  7. Exercise • What are the outcomes associated with each event?

  8. Event A and event B Event A and event B are mutually exclusive.

  9. Event C and Event D Event D Event C Event C and D are not mutually exclusive!

  10. Exercise • What are the probabilities for event A, B, C and D?

  11. Notation • In Set theory, a set is a collection of things. x B is a set. A is a subset of B, i.e. A A subset can also be empty. A null set, an empty set has no element. The notation of a null is .

  12. Intersection • The intersection of two sets C and D is the set of all elements which are contained in both C and D. Event D Event C

  13. Union • The union of sets C and D is the set of all elements that are either in C or in D, or in both. C

  14. Mutually Exclusive

  15. Quiz • Monitor three consecutive phone calls going through a telephone switching office. Class each one as a voice call (v) if someone is speaking, or a data (d) if the call is carrying a modem or fax signal. Your observation is a sequence of three letters each letter is either v or d. • Write down the following events: • First call is a voice call • One or more voice call

  16. Answer • Write down the following events: • First call is a voice call A1={vvv,vvd,vdv,vdd} • One or more voice call A4={vvv,vvd,vdv,dvv,vdd,dvd,ddv}

  17. Telephone Usage What is the sample space? What is the probability of the sample space?

  18. Telephone Usage What is the probability associated with the long call?

  19. Telephone Usage What is the probability associated with the brief call if you know the probability of a long call?

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