The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 12 General Probability Rules 10 10 06 Lecture 12 1 Review Outcome Sample space Event Union or Intersection and Complement Disjoint Venn diagram Basic rules For any event A P not A 1 P A If A and B are disjoint then P A B 0 For any two events A and B P A B P A P B P A B 10 10 06 Lecture 12 2 General Addition Rule 10 10 06 Lecture 12 3 Independence A and B are independent if knowing that one occurs does not change the probability that the other occurs For independent events A and B P A B P A P B Multiplication rule 10 10 06 Lecture 12 4 Cards Ex 1 A card is drawn from a deck of 52 playing cards What is the probability that the card is a club event A a king event B a club and a king event A B Are A and B independent 10 10 06 Lecture 12 5 Independent vs Disjoint If A and B are independent P A B P A P B If A and B are disjoint P A B 0 Note If P A 0 and P B 0 then disjoint A B are dependent A happens simply implies that B does not happen 10 10 06 Lecture 12 6 Conditional Probability The probability of an event measures how likely it will occur A conditional probability predicts how likely an event will occur under specified conditions P A B the conditional probability that A occurs uncertain given that B has occurred certain The condition contains partial knowledge 10 10 06 P A B P A B P B Lecture 12 7 Two way Table Ex 2 Prediction record of a TV weather forecaster over the past several years Forecast Sunny cloudy Rainy Row Sum Sunny 50 05 04 59 Actual Cloudy 04 10 02 16 Weather Rainy 10 05 10 25 Column Sum 64 2 16 1 How likely was the forecaster wrong What was the probability of rain What was the probability of rain given the forecast was sunny 10 10 06 Lecture 12 8 Gender of Children Ex 3 A family has two children Assume all four possible outcomes younger is a boy older is a girl are equally likely What is the probability that both are boys given that at least one is a boy Let A both are boys B at least one is a boy P A 1 4 P B 3 4 Why P A B P A B P B P A P B 1 3 10 10 06 Lecture 12 9 Stock Market Ex 4 A stock market analyst feels that the probability that a mutual fund company will get increased contributions from investors is 0 6 the probability becomes 0 9 if the stock market goes up the probability drops below 0 6 if the stock market drops with probability 0 5 the stock market rises The events of interest are A the stock market rises B the company receives increased contributions Calculate P A B and P A B 10 10 06 Lecture 12 10 Urn of Chips Ex 5 An urn contains 5 white chips and 4 blue chips Two chips are drawn sequentially without replacement What is the probability of obtaining the sequence white blue A 1st chip is white B 2nd chip is blue Want to know P A B Use the formula P A B P B A P A 10 10 06 Lecture 12 11 More Independence Conditions For any two events A and B P A B P A B P B P B A P A Two events A and B are independent If any of the following equivalent conditions holds i P A B P A ii P B A P B iii P A B P A P B 10 10 06 Lecture 12 12 Cards Ex 1 revisited A card is drawn from a deck What is the probability that the card is a club given the card is a king A the card is a king B the card is a club P A 4 52 P B A 1 52 Hence P B A P B A P A 1 4 Note that P B A P B This means knowing A occurs has no impact on the chance for occurrence of B In other words B A are independent 10 10 06 Lecture 12 13 High school athlete Ex 6 5 of male high school athletes go on to play at college level Of these 1 7 enter major league professional sports About 0 01 of the high school athletes who never compete in college enter professional sports A competes in college B competes professionally Q the probability that a high school athlete competes in college and then goes on to have a pro career Q a high school athlete goes to professional sports 10 10 06 Lecture 12 14 Tree Diagram Ex 6 continued 10 10 06 Lecture 12 15 Bayes Rule For any two events A and B P A B P A B P B P B P B A P B not A P B A P A P B not A P not A If P A and P B are not 0 or 1 then P A B P A B P B 10 10 06 Lecture 12 16 Quality Control Ex 7 A manufacturer is trying to find ways to reduce of defective parts The present procedure produces 5 defectives An inspection found that 40 of defectives and 15 of non defectives were produced by machine 1 Q What is the probability that a part produced by machine 1 is found to be defective Try a tree diagram 10 10 06 Lecture 12 17 Quality Control Ex 7 continued A a part is defective B a part is produced by machine 1 P A B P A 05 P not A 95 P B A 4 and P B not A 15 4 05 P A B 12 3 4 05 15 95 10 10 06 Lecture 12 18 Lab Test Ex 8 A lab test yields two possible results positive or negative 99 of people with a particular disease will produce a positive result But 2 of people without the disease will also produce a positive result Suppose that 0 1 of the population actually has the disease Q What is the probability that a person chosen at random actually has the disease given a positive result Try a tree diagram 10 10 06 Lecture 12 19 Lab Test Ex 8 continued D disease positive test result Want P D P D 001 P not D 0 999 P D 99 and P not D 02 Applying Bayes Rule 99 001 00099 P D 4 7 99 001 02 999 02097 10 10 06 Lecture 12 20 Bayes Rule general 10 10 06 Lecture 12 21 Take Home Message General Addition Rule Independence Conditional probability Multiplication Rule Bayes Rule Tree Diagram 10 …