Transcript More Probability - University of Rhode Island

More Probability
STA 220 – Lecture #6
1
Basic Probability Definition
• Probability of an event
– Calculated by dividing number of ways an event
can occur by number of total possible outcomes
# of ways
P(event ) 
total # of
2
Independence
• Two events A and B are independent if
knowing that one occurs
the probability that the other occurs.
– Example: Suppose event A is the event that you
toss a penny and it comes up heads and event B is
the event that you toss a nickel and it also comes
up heads
3
Dependence
• If one of the events influences the outcome of
the other than events A and B are
.
– Example: Suppose A is the event that it is snowing
outside and let B be the event that your shoes are
wet.
– Let A be the event that you do not attend a
difficult class and suppose event B is the
probability of getting an A
4
Independence and the Multiplication
Rule
• Rule 5: If A and B are independent,
P(A and B) = P(A)*P(B)
This is the multiplication rule for
5
Multiplication Rule
• Gregor Mendel used garden peas in some of the
experiments that revealed that inheritance
operates randomly. The seed color of Mendel’s
peas can be either green or yellow. Two parent
plants are “crossed” (one pollinates the other) to
produce seeds. Each parent plant carries two
genes for seed color, and each of these genes has
probability ½ of being passed to a seed. The two
genes that the seed receives, one from each
parent, determine its color: The parents
contribute their genes independently of each
other.
6
Multiplication Rule
• Suppose that both parents carry the G and the
Y genes. The seed will be green if both
parents contribute a G gene; otherwise it will
be yellow. If M is the event that the male
contributes a G gene and F is the event that
the female contributes a G gene, then the
probability of a green seed is
P(M and F)
= P(M)*P(F)
=
7
Multiplication Rule
• Sudden infant death syndrome (SIDS) causes babies to
die suddenly (often in their cribs) with no explanation.
Death from SIDS has been greatly reduced by placing
babies on their backs, but as yet no cause is known.
• When more than one SIDS death occurs in a family, the
parents are sometimes accused. One “expert witness”
popular with prosecutors in England told juries that
there is only a 1 in 73 million chance that two children
in the same family could have died naturally. The rate
of SIDS in a nonsmoking middle-class family is 1 in
8500
8
Multiplication Rule
• The prosecutor’s calculation was:
1
1
1
*

8500 8500 72,250,000
• Several women were convicted of murder on
this basis, without any direct evidence that
they harmed their children.
9
Caution!!
• 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
• Do not confuse disjointness with
independence. Disjoint events cannot be
10
AND vs OR
• If you want to compute the probability that 2
(or more) events both occur then you are
looking for the P(A and B)
– Keyword:
– If A and B are independent then recall
P(A and B) =
11
AND vs OR
• Example of “AND”
• Consider example of tossing a penny and
getting a heads and tossing a nickel and
getting a heads. Find the probability that both
coins come up heads:
P(Both coins show heads) =
P(Both coins show heads) = (0.5)*(0.5)
P(Both coins show heads) = 0.25 = 25%
12
AND vs OR
• If you want to find the probability that one
event occurs or another event occurs (or they
both occur) then you are looking for the P(A
or B)
• General addition rule says:
P(A or B) =
• Recall if A and B are disjoint events, then P(A
and B) = 0, so the addition rule becomes:
P(A or B) = P(A) + P(B)
13
AND vs OR Examples
• Suppose the probability of being left-handed
is 0.10 or 10%. In a family with two children,
what’s the probability of both children being
left-handed?
– Translate problem into shorthand:
P(Both left-handed) = P(1st is L and 2nd is L)
=
* P(2nd is L) = 0.10*0.10
P(Both left-handed) =
14
AND vs OR Examples
• Roll a die
– Let A be the event that you roll an even number =
{2,4,6}
– Let B be the event that you roll a number less than
or equal to 3 = {1,2,3}
– Find P(A or B)
P(A or B) =
P(A) = 3/6, P(B) = 3/6, P(A and B) = 1/6
P(A or B) = 3/6 + 3/6 – 1/6 =
15
Contingency Tables
• The following is a contingency table giving the
number of colleges in the US by region and
type:
Region\Type
Public
Private
Total
Northeast
266
555
821
Midwest
359
504
863
South
533
502
1035
West
313
242
555
Total
1471
1803
3274
16
Contingency Tables
• What is the total number of institutions of higher
education in the US?
–
• How many institutions are in the Midwest?
–
• How many institutions are public?
–
• How many institutions are private schools in the
south?
–
17
Contingency Tables
• Suppose we select one college at random.
What is the probability we select a college:
– In the midwest?
P(midwest) =
P(midwest) = 0.2636
– That is public?
P(public) = 1471/3274
P(public) =
18
Contingency Tables
• What’s the probability we select a public
college in the midwest?
– P(public AND in midwest) =
= 0.1097
– P(public OR in midwest)
= P(public) + P(midwest) –
= 0.4493 + 0.2636 – 0.1097
= 0.6032
19
Contingency Table
• What’s the probability we select a college in
the midwest or west?
– P(midwest OR west)
= P(midwest)+P(west)
= 0.2636 +
= 0.2636 + 0.1695
= 0.4331
20
Contingency Table
• Note that in all previous calculations the
denominator is always the
in the population
• The conditional probability P(A|B) means the
probability that A will occur given that B has
occurred.
– We are only interested in a certain column or row
of the table now, so the denominator will be the
total from that column or row
21
Contingency Table
– Example: What’s the probability a college is public
given that its in the northeast?
• ONLY interested in the northeast – denominator will be
• Of those colleges in the northeast, how many are also
public?
• P(public|northeast) =
= 0.3240
22
Contingency Tables
• What’s the probability a college is in the south
given that its private?
– Only interested in schools that are private
– What is denominator? 
– How many private schools are in the south?

– P(south|private) =
= 0.2784
23