Statistical Sciences 2244A/B Chapter Notes - Chapter 10: Venn Diagram, Conditional Probability, Sample Space
22 May 2018
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Department
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Stats 2244
Chapter 10
CHAPTER 10
Relationships among several events
- Now we are trying to describe the probability that both events A and B occur
- The sample space S is the rectangular area
- Events are areas within S which create a venn diagram
- The events in the image on the left are disjoint bc they dont overlap
- In the image on the right, the event {A and B} appear as the overlapping area that is common to
both A and B
- Ex: you want to study the next 2 single births at a local hospital
- You are counting girls, so 2 events of interest are:
- The events A and B are not disjoint (they are not mutually exclusive) – they occur together
whenever the next 2 single births at the hospital are baby girls
- We want to find the probability of the event {A and B} that BOTH babies are girls
- What is P(A and B)? ¼
o ½ x ½ = ¼
o This assumes that the 2nd baby still have a probability ½ of being a girl after the first one
was a girl
o The birth outcome of one couple is not influenced by the birth outcome of another
couple
o We say that the events girl on the first birth and give on the second birth are
independent
- Independence means that the outcome of the first event cannot influence the outcome of the
second event
- Multiplication rule for independent events
o Two events A and B are independent if knowing that one occurs does not change the
probability that the other occurs
o If A and B are independent, then P(A and B) = P(A)P(B)
o If this condition is not satisfied, then events A and B are dependent
o The multiplication rule also extends to collections of more than 2 events, provided that
all are independent
- Independence is often assumed in setting up a probability model when the events are describing
seem to have no logical connection
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- The multiplication rule P(A and B) = P(A)P(B) holds
- if A and B are independent, but not otherwise. The addition rule P(A or B) = P(A) + P(B) holds if
A and B are disjoint, but not otherwise
CHAPTER 10.2
Conditional Probability
- not all events are independent
- sometimes, the probability we assign to an evetn can change if we know that some other event is
true or has occurred
- when PA>0, the conditional probability of B, given A, is…
- The conditional probability P(B | A) makes no sense if the event A can never occur, so we
require that P(A) > 0 whenever we talk about P(B | A)
- Keep in mind the distinct roles of the events A and B in P(B | A) →Event A represents the
information we are given, and B is the event whose probability we are calculating
- Personal probability
o Special case of conditional probability
o Different persons may assign different personal probabilities to a given event bc their
knowledge and beliefs are different
CHAPTER 10.3
General Probability Rules
General Addition Rule
- If events A and B are disjoint events, then P(A or B) = P(A) + P(B)
- If events A and B are not disjoint events, they can occur together
o The probability that one or the other occurs is then less than the sum of their
probabilities
o Outcomes common to both are counted twice when we add probabilities, so we must
subtract this probability once
- General addition rule for any two events,
o For any two events A and B,
- If A and B are disjoint, the event {A and B} that both occur contains no outcomes and therefore
has probability 0
- So the general addition rule includes the addition rule for disjoint events
- The addition rule for disjoint events:
o )f events A, B, C, … are all disjoint in pairs, then Pat least one of these event occurs =
PA+PB+PC+…
- Venn diagrams:
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Now we are trying to describe the probability that both events a and b occur. The sample space s is the rectangular area. Events are areas within s which create a venn diagram. The events in the image on the left are disjoint bc they don(cid:495)t overlap. In the image on the right, the event {a and b} appear as the overlapping area that is common to both a and b. Ex: you want to study the next 2 single births at a local hospital. You are counting girls, so 2 events of interest are: The events a and b are not disjoint (they are not mutually exclusive) they occur together. We want to find the probability of the event {a and b} that both babies are girls whenever the next 2 single births at the hospital are baby girls. Independence means that the outcome of the first event cannot influence the outcome of the second event independent.
