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STAT131 Week 4 Lecture 1a Measuring Uncertainty

STAT131 Week 4 Lecture 1a Measuring Uncertainty. Anne Porter. How might we express this answer in terms of probability?. As the number of screws increases the probability of a failure increases. A small idea!. Video Clip Decisions Through Data , extract from Unit 1. Measuring Uncertainty.

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STAT131 Week 4 Lecture 1a Measuring Uncertainty

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  1. STAT131Week 4 Lecture 1aMeasuring Uncertainty Anne Porter

  2. How might we express this answer in terms of probability? As the number of screws increases the probability of a failure increases A small idea!

  3. Video Clip Decisions Through Data, extract from Unit 1.

  4. Measuring Uncertainty In an uncertain world many things are unpredictable • Will it rain tomorrow? • Will the St George Illawarra Dragons win next week? • Will there be an earthquake or lightening strike? In such situations, it is impossible to predict with certainty what will happen.

  5. Probability is used to quantify and describe precisely this unpredictability

  6. Provide a measure for these... ...Or if you can’t, decide how you would measure • The probability that the St George Illawarra Dragons will win their seventh game. • The probability that it will rain tomorrow • The probability that the train after the last STAT131 lecture to Sydney will be late. We can use subjective probability

  7. What is the probability of... • Getting a 6 when I toss this equal sided die? • Getting a 2 or a 3 when I toss this equal sided die? • Getting a 6 when I toss this unequal sided die? What thinking gave you these answers?

  8. What is the probability of... • Getting a 6 when I toss this equal sided die? • Getting a 2 or a 3 when I toss this equal sided die? • Getting a 6 when I toss this unequal sided die? What thinking gave you these answers? These are based on Equally Likely Outcomes

  9. What is the probability of a 5 with an unequally sided dice? Experiment Number or (Repetition) Relative frequency Value on die 1 2 3 4 5 6 7 8 9 : Very large n 5 2 3 1 6 4 4 3 : : : 1/1=1 1/2=0.5 1/3=0.33 1/4=0.25 1/5=0.2 1/6=0.17 : : : : :

  10. Plotting the relative frequency Relative frequency x 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 x x x x x x x x x x 1 2 3 4 5 6 7 8 9 10…………………large n Number of repetitions

  11. Measured by Probabilityis a measure of unpredictability Equally likely Outcomes Why assume equal likelihood? Subjective theoretical empirical I think Relative frequency Long run order

  12. P(event) = number of outcomes Total number of outcomes in the sample space P(event) = number of event occurs Total number of times the experiment is repeated Measuring Uncertainty • Subjective ‘I reckon the probability is…’ • Theoretical (equally likely) • Empirical (Long run frequency)

  13. Axiomatic probability Subjective Theoretical Empirical • If it is possible to measure with all methods then: • They will have the same numerical value • They will obey the same rules for combining probability

  14. Language and Notation • Experiment • Outcome • Sample Space • Null Event • Event (E) The observation of any phenomenon that is uncertain The outcome of an experiment that cannot be reduced to any simpler results (S) the collection of all outcomes {} or f is the collection of no outcomes Collection of outcomes

  15. Probability as a measure 1.00 • Event certain to happen • Event equally likely to happen as not • Event impossible to occur 0.5 0.00

  16. P(E) We write P(E) for the probability that event E will happen. So ... • P(I will die some day) = 1.0 • P(I will live forever) = 0.0 • P(Illawarra Dragons will make it to the top eight)= ?

  17. Gamblers fallacy • “If I toss a coin 100 times and get a head on each toss, I am almost certain to get a tail on the next toss” • What is wrong with this thinking? Coin is unfair and maybe the probability of a head is close to 1 Coin is fair and the probability of a head on the next toss is 0.5

  18. Probability of at least one of all events occurring? P(Sample space)= 1 S

  19. Nooverlap Events A and B are mutually exclusive P(A intersection B)=0 P(A and B)=0 S A B P(A happens or B happens)= P(A)+P(B)

  20. Events A and B are NOTmutually exclusive S overlap B A P(A happens or B happens)= P(A)+P(B)-P(A and B)

  21. Rules of Probability • Whichever type of probability is used, it must obey the rules of probability: 1. For any event E, 0 ≤ P(E) ≤ 1 2. P(S) = 1 (S = sample space) • P(A happens or B happens) = P(A) + P(B) ie P(AUB)=P(A)+P(B) if A and B are events with no outcomes in common (mutually exclusive events).

  22. Rules of probability 4. If A and B are NOT mutually exclusive then P(A happens of B happens)=P(A)+P(B)-P(A and B) 5. P(E occurs)+P(E does not occur)=1 Can be written in different notation

  23. Rules of Probability • A very useful consequence of these rules is: P(E occurs) = 1 - P(E does not occur) • Often the easiest way to find P(E occurs) is to find the probability that E does NOT occur and then to use this result.

  24. Rules of Probability 6. Independence Events A and B are independent if Or if

  25. Events A and B A Not A Total B 10 20 30 Not B 20 40 60 Total 30 60 90 Are events A and B independent.

  26. Events A and B A Not A Total B 10 20 30 Not B 20 40 60 Total 30 60 90 P(A)=30/90 P(B)=30/90 P(A).P(B)=30/90*30/90=1/9=P(A and B)=10/90 P(A|B)=10/30 =P(A) Therefore independence

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