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Introduction to Probability: Part I

Introduction to Probability: Part I. Definitions. An EXPERIMENT is an activity with observable results. For instance, if a quarter is tossed once, we can observe the results: heads or tails. An OUTCOME is any of the possible results of an experiment.

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Introduction to Probability: Part I

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  1. Introduction to Probability: Part I

  2. Definitions • An EXPERIMENT is an activity with observable results. • For instance, if a quarter is tossed once, we can observe the results: heads or tails. • AnOUTCOMEis any of the possible results of an experiment. • For instance, in the above experiment, both heads and tails are outcomes. • The SAMPLE SPACE of an experiment is the set of all possible outcomes of an experiment, usually denoted by S. • For instance, for the above experiment, S={heads, tails}.

  3. Example #1 • You have 3 light bulbs. Check them one at a time and classify each one as D for defective and G for good. Hence, if only the first 2 are defective, write DDG. • Find the sample space for this experiment.

  4. Remember the sample space is the set of all possible outcomes of the experiment. • The challenge is to make sure that all the possible outcomes are taken in account. • We take a systematic approach to finding all the outcomes. • We consider the following cases: all 3 bulbs are defective, 2 of the 3 are defective, 1 of the 3 is defective, and none is defective.

  5. Remark • It is noteworthy that we can easily find the number of elements in S without listing all the actual outcomes. • There are two possible outcomes for each bulb: D or G. • So, by the multiplication principle, there are (2)(2)(2)= 8 possible outcomes in S, as found earlier.

  6. Definition • An EVENT is any subset of the sample space of an experiment.

  7. Example #2 • You have 3 light bulbs. Check them one at a time and classify each one as D for defective and G for good. Hence, if only the first 2 are defective, write DDG. • For the above experiment, find the events: • E: "All the bulbs are defective." • F: "Exactly one bulb is defective." • G: "At least one of the bulbs is defective."

  8. Example #3 • Two distinguishable die are tossed. The sides that face up are observed. • Let S be the sample space. Find n(S). • Find n(E), n(F), and n(G) where • E is the event: “the sum of the sides is 5." • F is the event: “the sum of the sides is 12." • G is the event: “the sum of the sides is 1."

  9. Definition • The PROBABILITY of an event is a measure of the likelihood of the occurrence of an event. • When an event is very likely, the probability is close to 100%, i.e. 1. • When an event is very unlikely, the probability is close to 0%, i.e. 0.

  10. Example #4 • There are 27 employees in your office. Nine of them own an SUV. • If an employee is randomly selected, how likely is it that he owns an SUV?

  11. Formula for the Probability of an Event • Let E be an event in a sample space S. • If every outcome of S is equally likely, then the probability of the event E, denoted by p(E), is given by • Here n(E) and n(S) are the number of elements in E and S, respectively.

  12. Example #5 • You have 3 light bulbs. Check them one at a time and classify each one as D for defective and G for good. Hence, if only the first 2 are defective, write DDG. • Find the probability that a) All the bulbs that you checked are defective. b) Exactly 1 of the bulbs that you checked is defective.

  13. Example #6 • Two distinguishable die are tossed. The sides that face up are observed. • Find p(E), p(F), and p(G) where • E is the event: “the sum of the sides is 2." • F is the event: “the sum of the sides is 14." • G is the event: “the sum of the sides is greater 10."

  14. We will use the counting techniques that we have learned to compute probabilities.

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