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Probability Examples

Probability Examples. Jake Blanchard Spring 2010. Waste in Rivers (Example 2.21). A chemical plant dumps waste in two rivers (A and B). We can measure the contamination in these rivers. =event that A is contaminated =event that B is contaminated P()=0.2; P()=0.33; P()=0.1 (both)

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Probability Examples

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  1. Probability Examples Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers

  2. Waste in Rivers (Example 2.21) • A chemical plant dumps waste in two rivers (A and B). We can measure the contamination in these rivers. • =event that A is contaminated • =event that B is contaminated • P()=0.2; P()=0.33; P()=0.1 (both) • What is probability that at least one river is contaminated? Uncertainty Analysis for Engineers

  3. At least one is contaminated… • P()=P()+P()-P() • P()=0.2+0.33-0.1=0.43 • If B is contaminated, what is the probability that A is also contaminated? Uncertainty Analysis for Engineers

  4. If B, also A… • P(|)= P()/P() • P(|)=0.1/0.33=0.3 • What is the probability that exactly one river is polluted? Uncertainty Analysis for Engineers

  5. Exactly one polluted… • P()-P()=0.43-0.1=0.33 Uncertainty Analysis for Engineers

  6. Power Plants and Brownouts (Example 2.22) • We have 2 power plants (A and B) • =failure of A • =failure of B • P()=0.05; P()=0.07; P()=0.01 (both) • If one of the two plants fails, what is probability the other fails as well? Uncertainty Analysis for Engineers

  7. If one fails, the other fails as well… • P(|)= P()/P()=0.01/0.07=0.14 • P(|)= P()/P()=0.01/0.05=0.2 • What is the probability of a brownout (at least one fails)? Uncertainty Analysis for Engineers

  8. At least one fails… • P()=P()+P()-P() • P()=0.05+0.07-0.01=0.11 • What is probability that a brownout is caused by the failure of both plants? Uncertainty Analysis for Engineers

  9. Brownout from both failing… • Probability of brownout is 0.11 • Probability of both plants failing is 0.01 • Probability of brownout from both failing is 0.01/0.11=0.09 Uncertainty Analysis for Engineers

  10. A Useful Formula • P(D)=P(D|AX)*P(AX)+ P(D|AY)* P(AY)+ P(D|BX)* P(BX)+ P(D|BY)* P(BY) • This works as long as A, B and X, Y cover all possible outcomes Uncertainty Analysis for Engineers

  11. Highway Congestion (Example 2.26) I1 I3 I2 Uncertainty Analysis for Engineers

  12. Highway Congestion (Example 2.26) • E1=congestion on interstate 1 • P(E1)=0.1; P(E2)=0.2 • P(E1|E2)=0.4; P(E2|E1)=0.8 (Bayes’ Theorem) • P(E3|E1E2)=0.2; ie if no congestion on 1 or 2, then probability of congesion on 3 is 20% • Also, there is 100% probability of congestion on 1 if there is congestion on either 1 or 2 Uncertainty Analysis for Engineers

  13. Example 2.26 • There are 4 possibilities: E1E2, E1E2, E1E2, E1E2 • P(E1E2)=P(E1|E2)P(E2)=.4*.2=.08 • P(E1E2)=P(E2|E1)P(E1)=(1-.8) *.2=.02 • P(E1E2)=P(E1|E2)P(E2)=(1-.4)*.2=.12 • P(E1E2)=1-.08-.02-.12=.78 • P(E3)=P(E3|E1E2)P(E1E2)+ P(E3|E1E2)P(E1E2)+ P(E3|E1E2)P(E1E2)+ P(E3|E1E2)P(E1E2)=0.376 Uncertainty Analysis for Engineers

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