Transcript Key Concepts of the Probability Unit

Key Concepts of the Probability Unit
 Simulation
 Probability rules
 Counting and tree diagrams
 Intersection (“and”): the multiplication rule, and independent
events
 Union (“or”): the addition rule, and disjoint events
 Venn diagrams
 Conditional probability and Bayes Rule
Simulation
 Can often be used to estimate probabilities, especially when
there is a complex series of events
 Is a valid technique for verifying the results of a probability
model
 Is accepted on the AP Exam
 Can be done using a calculator, computer, or random number
table
Counting
 It is necessary to determine how many outcomes are in a
sample space before we can determine the probability of an
event
 Usually requires determining how many ways each part of an
event can happen, then finding the product of these
 Counting problems usually involve combinations and
permutations, concepts that are (surprisingly) not covered in
this book
Tree Diagrams
 Very useful for illustrating and determining how many ways
outcomes can occur (how many items are in a sample space)
 Can also be used to calculate the associated probability of
each outcome
Intersection
 The intersection of P(A) and P(B), means the probability of
both A and B occurring, and is denoted by P( A  B)
 If the outcome of event A has no impact upon the outcome of
event B, they are said to be independent. Calculating P( A  B)
then is very easy, it is just P(A) x P(B).
 Example: probability of rolling a “6” on a die, then drawing a “red” card.
 If the outcome of event A has an impact upon the outcome of
event B, they are said to be dependent. Calculating P( A  B)
then is more involved: it is P(A) x P(B/A), read as Probability of B
given A.
 Example: probability of drawing a red card, then drawing another red
card/given that the first card was red
Union
 The union of P(A) and P(B), means the probability of A or B
occurring, and is denoted by P( A  B)
 If the outcome of event A has no possibility of occurring at the
same time as event B, they are said to be disjoint. Calculating
P( A  B) then is very easy, it is just P(A) + P(B).
 Example: probability of rolling a “6” on a die or rolling a “3”.
 If the outcome of event A can occur at the same time as event B,
they are said to be not disjoint. Calculating P( A  B)
then is more involved, it is P(A) + P(B) – P( A  B)
 Example: probability of rolling a “greater than 3” on a die or rolling an “even
number”: P(greater than 3) + P(even) – P(4 or 6)
Venn Diagrams
 Very useful for Intersection and Union problems
 Visual displays of Intersection, Union, and Complementary
probabilities
Re
Remember that P(D)
is equal to the sum
of the light green
and blue regions!
P(D) is equal to the sum of
the light green and blue
regions!
Conditional Probability
 Conditional probabilities are a logical next step from the
Conditional Distributions we studied in 4.2
 Can be calculated from unconditional probabilities using this
formula:
P( A and B)
P( B / A) 
P( A)
 Example: P(Draw a red card 2nd, given a red card was drawn
1st ) is equal to P(red card 1st x red card 2nd)/P(red card 1st),
which equals 13 12

52


51
1352
Example of Conditional Probability
Bayes Rule
 Bayes rule allows us to calculate P(B/A) if we know P(A/B)
 Often it is easier to derive P(B/A) without using Bayes Rule
by using a Tree Diagram (see textbook Ex. 6.31)
 Bayes Rule:
P( A / B) P( B)
P( B / A) 
P( A)
Example of Bayes Rule
 From our previous example, we saw that P(“A”/liberal arts) was
34%. Can we use the information we have to find P(liberal
arts/“A”)? Recall that…
 So, P(lib arts/A) = P(A/lib arts)P(lib arts)
•
P(A)
(.34)(.63)/.34 = .6314