狠狠撸

狠狠撸Share a Scribd company logo

Conditional probability

?Download as PPTX, PDF?
?
S
suncil0071

This document discusses conditional probability and independence. It defines conditional probability as P(B|A), the probability of event B given that event A has occurred. To calculate this, one considers only the outcomes where A occurred and calculates the fraction where B also occurred. Two events A and B are independent if P(B|A) = P(B), meaning the probability of B is unaffected by the occurrence of A. The general multiplication rule for any events A and B is P(A and B) = P(A) × P(B|A) or P(A) × P(B|A). Disjoint events cannot be independent as the occurrence of one rules out the other. Conditional probabilities are best understood

1 of 9
Downloaded 683 times
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario Probability Conditional Probability
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 2 Conditional Probability ? When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.” ? A probability that takes into account a given condition is called a conditional probability.
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 3 Conditional Probability (cont.) ? To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find in what fraction of those outcomes B also occurred. ? Note: P(A) cannot equal 0, since we know that A has occurred. (A and B)( | ) ( ) PP P ?B A A
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 4 Independence ? Independence of two events means that the outcome of one event does not influence the probability of the other. ? With our new notation for conditional probabilities, we can now formalize this definition: ? Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 5 The General Multiplication Rule ? When two events A and B are independent, we can use the multiplication rule for independent events : P(A and B) = P(A) x P(B) ? However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 6 The General Multiplication Rule (cont.) ? We encountered the general multiplication rule in the form of conditional probability. ? Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule: ? For any two events A and B, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B)
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 7 Independent ≠ Disjoint ? Disjoint events cannot be independent! Well, why not? ? Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. ? Thus, the probability of the second occurring changed based on our knowledge that the first occurred. ? It follows, then, that the two events are not independent. ? A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake.
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 8 Depending on Independence ? It’s much easier to think about independent events than to deal with conditional probabilities. ? It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities. ? Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.
Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 9 Tables and Conditional Probability ? It is easy to understand conditional probabilities with contingency tables

More Related Content

Conditional probability

  • 1. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario Probability Conditional Probability
  • 2. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 2 Conditional Probability ? When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.” ? A probability that takes into account a given condition is called a conditional probability.
  • 3. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 3 Conditional Probability (cont.) ? To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find in what fraction of those outcomes B also occurred. ? Note: P(A) cannot equal 0, since we know that A has occurred. (A and B)( | ) ( ) PP P ?B A A
  • 4. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 4 Independence ? Independence of two events means that the outcome of one event does not influence the probability of the other. ? With our new notation for conditional probabilities, we can now formalize this definition: ? Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)
  • 5. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 5 The General Multiplication Rule ? When two events A and B are independent, we can use the multiplication rule for independent events : P(A and B) = P(A) x P(B) ? However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.
  • 6. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 6 The General Multiplication Rule (cont.) ? We encountered the general multiplication rule in the form of conditional probability. ? Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule: ? For any two events A and B, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B)
  • 7. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 7 Independent ≠ Disjoint ? Disjoint events cannot be independent! Well, why not? ? Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. ? Thus, the probability of the second occurring changed based on our knowledge that the first occurred. ? It follows, then, that the two events are not independent. ? A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake.
  • 8. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 8 Depending on Independence ? It’s much easier to think about independent events than to deal with conditional probabilities. ? It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities. ? Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.
  • 9. Copyright ? 2012 Pearson Canada Inc., Toronto, Ontario 狠狠撸 15- 9 Tables and Conditional Probability ? It is easy to understand conditional probabilities with contingency tables