1 / 16

AEN 202 Biosystems Engineering Problems

AEN 202 Biosystems Engineering Problems. Conditional Probability. Symbol for this: P(A|B). What is it? Probability of event A, given that event B has already occurred. Example: getting a cold is more likely if someone around us has a cold!. General Multiplication Rule.

felton
Download Presentation

AEN 202 Biosystems Engineering Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. AEN 202 Biosystems Engineering Problems

  2. Conditional Probability Symbol for this: P(A|B) What is it? Probability of event A, given that event B has already occurred. Example: getting a cold is more likely if someone around us has a cold!

  3. General Multiplication Rule If A,B are independent we know: If A,B are NOT independent we use conditional probability in the multiplication rule:

  4. Baye’s Thereom What is it? Inverse problem - if a conditional event has occurred, what is the probability it was caused by a particular agent? Example: 3 suppliers of transformers. If we have a bad transformer, what is the probability it came from supplier number 1?

  5. Baye’s Thereom We need first the TOTAL PROBABILITY of an event that can depend on other events: P(A) = P(B1)P(A|B1) + … + P(Bn)P(A|Bn) How to compute? This is the overall likelihood that a conditional event will happen.

  6. An Example Given: P(B1)=0.7 and P(B2)=0.3, are the amounts blue M&M’s supplied by two companies. Supplier 1 has a 5% probability of placing a cracked M&M in a 1lb bag; Supplier 2 has a 10% probability. Required: What is the probability of finding a cracked M&M in a single bag? Answer: this is a Total Probability problem. Note that P(B1|A)=0.05 and P(B2|A)=0.10. Thus: P(A)=0.7(0.05) + 0.3(0.10) = 0.035+0.03 = 0.065 or 6.5% likelihood of a broken candy in a given bag.

  7. Baye’s Thereom What is it? Inverse problem - if a conditional event has occurred, what is the probability it was caused by a particular agent? Example: If we have a cracked M&M in a bag, what is the probability it came from supplier number 1?

  8. Baye’s Thereom P(B1|A) = P(B1)P(A|B1) /P(A) How to compute? We use the TOTAL PROBABILITY in the denominator: This is the likelihood that a conditional event (an effect, A) arose from a particular agent (cause, B1).

  9. Return to the Example Given: P(B1)=0.7 and P(B2)=0.3, are the amounts blue M&M’s supplied by two companies. Supplier 1 has a 5% probability of placing a cracked M&M in a 1lb bag; Supplier 2 has a 10% probability. Required: What is the probability that a cracked M&M came from supplier B2? Answer: Note that the total probability of a cracked M&M is P(A)=0.065 from our previous solution. Thus: P(B2|A)=0.3(0.1)/0.065 = 0.4615 or 46.5% likelihood of a broken candy from supplier 2, despite the fact only 30% of candy comes from them.

  10. Mathematical Expectation Example: You buy tires rated for 60,000 miles. This is the expected life of the tires. What is it? An average. Of course, your experience may differ depending on driving habits and random chance.

  11. Mathematical Expectation Definition: E = a1p1 + a2p2 + … + anpn What is it used for? Typically, as a tool for decision making. The ai’s are values of individual outcomes. The pi’s are probabilities of those outcomes.

  12. An (contrived) Example Given: Bet on a 4-horse race, with pay-off $100 if Horse 1 wins and $50 if horse 2 wins. Nothing if horses 3 or 4 win. What is the expected value of your bet? Note: a1 = $100, a2 = $50, a3=a4=$0 p1 = p2 = p3 = p4 = 1/4 (assumed) Answer: E = $100(1/4) + $50(1/4) + $0(1/4) + $0(1/4) = $25 + $12.50 = $37.50 an unusually good bet!

  13. Mathematical Expectation See the examples in text for more realistic uses! It is a powerful technique.

  14. MidTerm • 100 points • 20% course grade • open-book, closed notes • bring calculator • 9 problems, through chapter 3 • covers material in quizzes, homework and lectures

  15. MidTerm • Descriptive statistics: definitions & calculations • 45 points • Set operators and Venn diagrams • 9 points • Probability (incl. Conditional probability & Bayes Thereom) • 36 points • Mathematical Expectation • 10 points

  16. Biosystems & Ag.Engineering

More Related