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The General Addition Rule

- When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14:

P(A B) = P(A) + P(B)

- However, when our events are not disjoint (not mutually exclusive), this earlier addition rule will double count the probability of bothA and B occurring. Thus, we need the General Addition Rule.
- Let’s look at a picture…

The General Addition Rule

- General Addition Rule:
- For any two events A and B,

P(A B) = P(A) + P(B) – P(A B)

- The following Venn diagram shows a situation in which we would use the general addition rule:

Addition Rule Example:

- A random sample of 250 working adults found that 37% access the Internet at work. 44% access the Internet from home. And 21% Access the Internet at both work and home. What is the probability that a person in this sample will access the Internet from home or at work?

Addition Rule Example 2:

Below is a Venn Diagram of a sample of 200 women who like yoga to relieve stress and who like running to relieve stress.

- What is the probability that a woman likes running?
- What is the probability that a woman likes running, but doesn’t like yoga?
- What is the probability that a woman doesn’t like yoga and doesn’t like running?
- What is the probability that a woman will like running or yoga, but not both?

0.12

conditional PROBABILITIES

- Back in Chapter 3, we looked at contingency tables and talked about conditional distributions.
- When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of BgivenA.”
- A probability that takes into account a given condition is called a conditionalprobability.

conditional PROBABILITIES (CONT.)

- To find the probability of the event Bgiven the event A, we restrict our attention to the total outcomes in A. We then find the fraction of those outcomes in B that also occurred.
- Note: P(A) cannot equal 0, since we know that A has occurred.

Conditional Probability Example:

- A random sample of 250 working adults found that 37% access the Internet at work. 44% access the Internet from home. And 21% Access the Internet at both work and home. What is the probability that an adult accesses the internet at home, given they access it at work?
- Tell: ____% of adults who access the internet at work also access it at home.

The General Multiplication Rule

- When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14:

P(A B) =P(A) xP(B)

- However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the GeneralMultiplicationRule.

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 B) = P(A) P(B|A)

or

P(A B) = P(B) P(A|B)

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).)

Independence Example:

- Earlier it was mentioned that 37% of working adults access the Internet at work. 44% access the Internet from home. And 21% Access the Internet at both work and home.
- Are accessing the internet from work and accessing the internet from work independent? Are they disjoint?

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.
- Consider 2 disjoint events: getting an A in this course and getting a B in this course. These are disjoint because they have nothing in common. However, if you did earn an A in the course, does the affect your probability of earning a B? YES! Your probability of earning a B in the course is now 0. It cannot happen.
- 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.

Examples: Independence vs. Disjoint

- The AAPOR is an association of about 1600 individuals who share an interest in public opinion and survey research. They report that typically as few as 10% of random phone calls result in a completed interview. Reasons are varied, but some of the most common include no answer, refusal to cooperate, and failure to complete the call.
- Which of the following events are disjoint, independent, or neither?
- A = Your phone # is randomly selected.

B = You’re not at home at dinnertime when they call.

- A = As a selected subject, you complete the interview.

B = As a selected subject, you refuse to cooperate.

- A = You are not at home when they call at 11 AM.

B = You are employed full time.

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.

Tables and Conditional Probabilities

- It is much easier to see conditional probabilities using contingency tables:
- Find the probability that a student is a male.
- Find the probability that a student is not a nursing major.
- Find the probability that a student is a female, given the student is a nursing major.
- Find the probability the student is male or a nursing major.
- Find the probability the student is female and not a nursing major.

Create a Contingency Table

- Create a contingency table for the following samples of textbook pages:
- 48% of pages had a data display
- 27% of pages had an equation
- 7% had both a data display and an equation

Using the Table Determine the Following:

- What is the probability that a randomly selected sample page with an equation also had a data display?
- Are having an equation and have a data display disjoint events?
- Are having an equation and having a data display independent events?

Drawing Without Replacement

- Sampling withoutreplacementmeans that once one individual is drawn it doesn’t go back into the pool.
- We often sample without replacement, which doesn’t matter too much when we are dealing with a large population.
- However, when drawing from a small population, we need to take note and adjust probabilities accordingly.
- Drawing without replacement is just another instance of working with conditional probabilities.

Drawing Without Replacement Example

- In a box of cupcakes, there are five cupcakes with red icing, 4 with yellow icing, and 3 with green icing. If 2 of the cupcakes are randomly selected from the box (no peeking to see which color ), what is the probability that the first cupcake has red icing and the second cupcake has green icing? What about the probability that the first has red icing and the second has red icing? (The first cupcake is not replaced before the second cupcake is selected).

Tree Diagrams

- A treediagramhelps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree.
- Making a tree diagram for situations with conditional probabilities is consistent with our “make a picture” mantra.

Tree Diagrams (cont.)

- Figure 15.5 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together.
- All the final outcomes are disjoint and must add up to one.
- We can add the final probabilities to find probabilities of compound events.

Example Using Tree Diagram

- What is the probability that a randomly selected student will be a binge drinker who has an alcohol related car accident?
- What is the probability that a selected student has had an alcohol related car accident?

Reversing the Conditioning

- Reversing the conditioning of two events is rarely intuitive.
- Suppose we want to know P(A|B), and we know only P(A), P(B), and P(B|A).
- We also know P(AB), since

P(A B) = P(A) x P(B|A)

- From this information, we can find P(A|B):

What Can Go Wrong?

- Don’t use a simple probability rule where a general rule is appropriate:
- Don’t assume that two events are independent or disjoint without checking that they are.
- Don’t find probabilities for samples drawn without replacement as if they had been drawn with replacement.
- Don’t reverse conditioning naively.
- Don’t confuse “disjoint” with “independent.”

recap

- The probability rules from Chapter 14 only work in special cases—when events are disjoint or independent.
- We now know the General Addition Rule and General Multiplication Rule.
- We also know about conditional probabilities and that reversing the conditioning can give surprising results.

Recap (cont.)

- Venn diagrams, tables, and tree diagrams help organize our thinking about probabilities.
- We now know more about independence—a sound understanding of independence will be important throughout the rest of this course.

Assignments: pp. 361 – 365

- Day 1: # 1, 9, 13, 19, 31
- Day 2: # 2, 6, 8, 12, 15, 23, 27, 33, 35
- Day 3: # 4, 10, 16, 17, 21, 25, 29, 41

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