Chapter 15 Probability Rules!
The General Addition Rule: For any two events A and B, the probability of A or B is
Example: (page 343) A survey of college students found that 56% live in a campus
residence hall, 62% participate in a campus meal program, and 42% do both.
What is the probability that a randomly selected student either lives or eats on campus?
Example: (page 344) Based on the Venn diagram, what is the probability that a
randomly selected student
a) lives off campus and doesn’t have a meal program?
b) lives in a residence hall but doesn’t have a meal program?
Example: (Page 344) In this book it has been found that:
48% of the pages have some kind of data display
27% of the pages have an equation
7% of the pages have both a data display and an equation
a) Display these results in a Venn diagram
b) What is the probability that a randomly selected page has neither a data display not an
equation?
c) What is the probability that a randomly selected page has a data display but no
equation?
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Conditional Probability: The probability of B given A is
* P(A) cannot equal 0, since we know that A has occurred.
Example: (Page 348) A survey of college students found that 56% live in a campus
residence hall, 62% participate in a campus meal program, and 42% do both.
While dining in a campus facility open only to students with meal plans, you meet
someone interesting. What is the probability that your new acquaintance lives on
campus?
The General Multiplication Rule
For any two events A and B, the probability of A and B is
The general multiplication rule for compound events does not require the events to be
independent.
We can also write
When events A and B are independent
Independence: Events A and B are independent whenever
Example: (Page 349) From the previous example, our survey told us that 56% of college
students live on campus, 62% have a campus meal program, and 42% both. Are living
on campus and having a meal plan independent?
Are they disjoint?
Disjoint events cannot be independent.
For example, __________________________________________. These events are
disjoint because they have no common outcomes.
What is the probability of getting a B, given that you have learned you got an A?
So these events __________________________________________.
____________________________ events cannot be independent.
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We cannot apply ________________________________ to independent events.
Example: (Page 350) The American Association for Public Research (AAPOR) is an
association of about 1,600 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 independent, which are disjoint, and which are
neither?
a) A= Your telephone number is randomly selected.
B = You’re not at home at dinnertime when they call.
b) A = As a selected subject, you complete the interview.
B = As a selected subject, you refuse to cooperate.
c) A= You are not at home when they call at 11 a.m.
B = You are employed full-time.
Example: (Page 353) From a previous example:
48% of the pages have some kind of data display
27% of the pages have an equation
7% of the pages have both a data display and an equation
a) Make a contingency table for the variables display and equation.
Equation
Yes
No
Total
Yes
No
Total
b) What is the probability that a randomly selected page with an equation also had a data
display?
c) Are having an equation and having a data display disjoint events?
d) Are having an equation and having a data display independent events?
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Drawing without replacement:
For example, pulling a card out of a deck and keeping it.
Tree Diagram:
-In a tree diagram, we ________________ the probabilities of the branches together.
-All the final outcomes ____________________________________________________.
-We can add the final probabilities to _________________________________________.
Reversing the Conditioning
Suppose we want to know P(A|B), and we know only P(A), P(B), and P(B|A).
We also know P(A ∩ B), since
From this information, we can find P(A|B):
Bayes’s Rule: When we have P(A|B) and we want to find the reverse probability P(B|A)
we need to find P(A∩B) and P(A). We can use a tree diagram to help, or Bayes’s Rule:
Example: (Page 359) A recent Maryland highway safety study found that in 77% of all
accidents the driver was wearing a seatbelt. Accident reports indicated that 92% of those
drivers escaped serious injury (defined as hospitalization or death), but 63% of the nonbelted drivers were so fortunate.
Use a tree diagram to find the probability that a driver who was seriously injured wasn’t
wearing a seatbelt.
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