Download 5 Minute Check, 26 Sep

Prob and Stats, Oct 16
Probability – Using a Two-Way Table to
Compute Probability Three Ways
Book Sections: N/A
Essential Questions: How can I compute the probability of any event?
How can I use a table to compute simple probability, compound
event probability and conditional probability?
Standards: PS.SPCR.1, .1a, .1b, .1c, .6, .7, PS.SPMJ.2
Probability
• Probability is, was, and always will be a number between
0 and 1 (inclusive).
The Two-Way Table
• A two-way table is a display of connected data, which is
displayed in rows and columns
• The connections of rows and columns define data
relationships
 There are at least two connected relationships woven into every
2-way table
Example
The following Venn diagram and 2-way table show the
relationship of a group of high school students who either play a
sport, study a foreign language, or do both or neither.
The First Thing (if required)
• The very first thing to always check for in a 2-way table
is that it is totaled (rows and column added) and that you
have a grand total for the table.
• If it is not done, make that your FIRST STEP in this
process.
Using the Two-Way Table
The simple probability computation
P(event) =
Pertinent row or column total
Table grand total
• The probability of an event is the event row or column total / table
grand total
Examples
On the first day of season practice, a high school’s football
players’ hair and eye color were noted. The results are
summarized below. Compute each probability:
Simple Probability - Experimental
On the first day of season practice, a high school’s football
players’ hair and eye color were noted. The results are
summarized below. Compute each probability:
Compute the following probabilities for a single, randomly
chosen player:
P(green eyes)
P(brown hair)
Mutually Exclusive Events
• Events A and B are mutually exclusive if they cannot occur
at the same time.
• Events A and B are inclusive if there is some (common)
outcome that can be both events.
Mutually Exclusive Events
• Events A and B are mutually exclusive if they cannot occur
at the same time.
Sample Space
A
B
Not occurring simultaneously means there is no overlap between the
events
Not Mutually Exclusive
• The opposite of mutually exclusive is ‘inclusive’ – which
means events can occur at the same time.
• We can say inclusive or not mutually exclusive to
describe this phenomenon.
Inclusive Events
• Events A and B are not mutually exclusive if they can
occur at the same time.
Sample Space
A∩B
A
B
The overlap (green section) is A ∩ B, also known as A and B
Compound “OR” Probability
• If two events, A and B are mutually exclusive, then
P(A or B) = P(A) + P(B)
• If two events, A and B are not mutually exclusive, then
P(A or B) = P(A) + (B) – P(A and B)
Where (A and B) are outcomes of both
A and B at the same time
Reminder
• To use a two-way table, you must have row and column
totals, and a table grand-total
• The very first thing to always check for in a 2-way table
is that it is totaled (rows and column added) and that you
have a grand total for the table.
• If it is not done, do this FIRST.
• If has been done, do not do it again.
The Two-Way Table
• A two-way table is a display of connected data, which is
displayed in rows and columns
• The connections of rows and columns define data
relationships
 Row to row or column to column relationships are mutually
exclusive
 Row to column relationships are inclusive
 Row and column intersections display the value of set
intersections
Table Example
There following table summarizes the healthcare choice of
the 386 employees at the Titan Corporation and whether or
not they have dependent children in their care.
Dep Child
No Child
HMO
145
39
BC/BS
85
42
None
23
52
Table Example
Compute the following probabilities:
P(HMO or have dep children)
P(HMO or have no coverage)
Dep Child
No Child
Total
HMO
145
39
184
BC/BS
85
42
127
None
23
52
75
Total
253
133
386
Using a Two-Way Table
To Compute Compound Event Probability
• Computing P(A or B)
0. If not a given, sum every row and column (including
your new totals row and column) to compute every event
total and a ‘grand total’
1. Find event A on row or column heading, scribe a line
2. Find event B on row or column heading, scribe a line
3. If lines are parallel – add row or column totals, put
answer over grand total, simplify (ME probability)
4. If lines cross, add row and column totals, subtract ‘cross
value’ put answer over grand total, simplify (I probability)
Read the Problem  Decide
• Upon reading any problem that contains the word or, you
must
 Decide if the two events are mutually exclusive or inclusive
 If inclusive, figure the points of overlap and find that
probability
 Apply the correct addition rule form that applies to the problem
• The table lines will guide you in these decisions
Examples
On the first day of season practice, a high school’s football
players’ hair and eye color were noted. The results are
summarized below. Compute each probability:
P(red hair or green eyes)
P(black or brown hair)
P(blond hair or hazel eyes)
Conditional Probability and the
Two-Way Table
• If we fix a row or column with a condition, we
can use it to limit the computation to that
condition. When using this tool, always total the
table before doing anything else.
Computing Within a Condition
Within the condition row or column
P(event) =
Pertinent entry
Row or Column total
• The probability of a condition occurs within a fixed row or
column, and is the event value (within the condition) / row or
column total
Examples
The following table shows the marital status of adults over 18
years of age at the 2010 census. All numbers are x 1 million:
Never married
Married
Widowed
Divorced
Males
28.6
62.1
2.7
9.0
Females
23.3
62.1
11.3
12.7
Compute each probability: One adult is selected randomly
P(widowed)
P(male)
Given that the person selected is a male, what is P(married)
Given that the person selected is divorced, what is P(female)
Examples
The following table shows the marital status of adults over 18
years of age at the 2010 census. All numbers are x 1 million:
Never married
Married
Widowed
Divorced
Males
28.6
62.1
2.7
9.0
102.4
Females
23.3
62.1
11.3
12.7
109.4
51.9
124.2
14.0
21.7
211.8
Compute each probability: One adult is selected randomly
P(widowed)
P(male)
Given that the person selected is a male, what is P(married)
Given that the person selected is divorced, what is P(female)
Examples
On the first day of season practice, high school football
players’ hair and eye color were noted. The results are
summarized below. One player is randomly chosen, compute
each probability:
Given that the player has blond hair, P(blue eyes)
Given that the player has hazel eyes, P(red hair)
Given that the player has brown eyes, P(brown hair)
Three Ways of Using a Table
• How do you know what is being asked – Read the
problem
 If a simple probability statement – P(event) find the event total
and divide that entry by grand total
 If compound event probability, P(a or b), draw lines, determine
case add and subtract (if applicable), divide by grand total
 If conditional probability, you can tell by the word given –
Given a, find P(b) – work within row or column of a, find b,
divide by row or column total
• No matter what, total the table if not already done so
Example
A student is randomly chosen: Compute:
P(does not play a sport)
P(does not play a sport or does not take a language)
Given that the student plays a sport, P(takes a language)
Class work: CW 10/16/15, 1-5
Homework: Due 10/19/15, 1-2