Download Conditional Probability Earlier we learned how to solve probability

Conditional Probability
Earlier we learned how to solve probability problems involving the rolling of 2 dice.
Example
If E is the event that the sum is 5 when two fair dice are rolled, what is P(E)?
To solve this, we first construct the equally-likely sample space, S (with 36 elements)
11
21
31
41
51
61
12
22
32
42
52
62
13
23
33
43
53
63
14
24
34
44
54
64
Next, by the Basic Probability Principle( )
( )
( )
15
25
35
45
55
65
16
26
36
46
56
66
Conditional Probability
What if I tell you that I have already peeked and the sum is odd? Does that matter?
Example
If E is the event that the sum is 5 when two fair dice are rolled, what is P(E)?
To solve this, we first construct the equally-likely sample space, S (with 36 elements)
11
21
31
41
51
61
12
22
32
42
52
62
13
23
33
43
53
63
14
24
34
44
54
64
Next, by the Basic Probability Principle( )
( )
( )
15
25
35
45
55
65
16
26
36
46
56
66
Conditional Probability
(
)
(
(
)
)
When you have a probability which includes an extra condition
(here the extra condition was that the sum was odd), that probability is called a
CONDITIONAL PROBABILITY and we have a special notation for it.
Suppose we call the condition, F. F = “the sum is odd”
(
(
)
(
)
(
).
) is called a conditional probability and is read ‘ the probability of E given F ’
We could also just write (
).
Conditional Probability
This leads us to the formal definition of conditional probability:
DEFINITION:
When we compute the probability of event E assuming that the event F has already
). We
occurred, we call this the conditional probability of E given F, denoted, (
read this as “the probability of E given F.”
Conditional Probability
We can use Venn Diagrams to examine this idea more closely.
n(E∩F)
n(E)
n(F)
By the Basic Probability Principle (and assuming an equally-likely sample space):
(
)
(
)
( )
and
(
)
(
)
( )
Conditional Probability
We can use Venn Diagrams to examine this idea more closely.
n(E∩F)
n(E)
n(F)
…and if we divide both the numerator and denominator of the two fractions above by
the number of elements, n(S) in the sample space, we can rewrite this as:
(
)
(
)
( )
(
)
( )
and
(
)
(
)
( )
(
)
( )
Conditional Probability
DEFINITION:
When we compute the probability of event E assuming that the event F has already
). We
occurred, we call this the conditional probability of E given F, denoted, (
read this as “the probability of E given F.”
(
)
(
)
( )
(
)
( )
and
(
)
(
)
(
( )
)
( )
When dealing with equally-likely sample spaces, it will probably be easier to use the
first way of computing conditional probabilities than the second way.
In other words, most of the time, you will find it easier just to use
(
)
(
)
( )
.
Conditional Probability
When dealing with equally-likely sample spaces, it will probably be easier to use the
first way of writing conditional probabilities than the second way.
In other words, most of the time, you will find it easier just to use
(
)
(
)
( )
.
Example:
If two cards are drawn without replacement from an ordinary deck of cards, what is the
probability that the second card is a heart given that the first card was a heart?
Conditional Probability
When dealing with equally-likely sample spaces, it will probably be easier to use the
first way of writing conditional probabilities than the second way.
In other words, most of the time, you will find it easier just to use
(
)
(
)
( )
.
Example:
A container has 35 green marbles, 20 blue marbles and 4 red marbles. Two marbles
are randomly selected without replacement. If E is the event that a green token is
selected first and F is the event that the second token is not green, compute
(
).
Conditional Probability
Example:
The graph has stats about distracted drivers below the age of 40 who were involved in
fatal crashes. If one driver is selected from the 2767 tallied and if a driver in this group
is under 20, what is the probability of the accident being related to cell phone use?
Product Rule for Probabilities
Product Rule for Probabilities:
(
)
( )
(
)
This formula allows us to compute the probability of the intersection of two events (the
probability of event E AND event F).
Example:
A container has 35 green marbles, 20 blue marbles and 4 red marbles. Two marbles
are randomly selected without replacement. What is the probability that the first one is
blue and the second one is red?
Utilizing Tree Diagrams for Probability Calculations
Brianna is in a lottery for a dorm room in one of two new campus dorms. She will
randomly draw a card to get space in either Xavier or Young (dorms). She will also
draw a card to determine if she gets a room or apartment. The number of cards in
each case is proportional to the number of availabilities. 30% of the available spaces
are in Xavier, 80% of the available spaces in Xavier are apartments and 40% of the
spaces in Young are apartments.
Utilizing Tree Diagrams for Probability Calculations
Brianna is in a lottery for a dorm room in one of two new campus dorms. She will
randomly draw a card to get space in either Xavier or Young (dorms). She will also
draw a card to determine if she gets a room or apartment. The number of cards in
each case is proportional to the number of availabilities. 30% of the available spaces
are in Xavier, 80% of the available spaces in Xavier are apartments and 40% of the
spaces in Young are apartments.
It is easy to summarize this information
in a tree diagram.
What is the probability that Brianna gets
an apartment in Young (Young ∩ Apt) ?
Utilizing Tree Diagrams for Probability Calculations
Brianna is in a lottery for a dorm room in one of two new campus dorms. She will
randomly draw a card to get space in either Xavier or Young (dorms). She will also
draw a card to determine if she gets a room or apartment. The number of cards in
each case is proportional to the number of availabilities. 30% of the available spaces
are in Xavier, 80% of the available spaces in Xavier are apartments and 40% of the
spaces in Young are apartments.
It is easy to summarize this information
in a tree diagram.
What is the probability that Brianna gets
housed in Xavier?
Utilizing Tree Diagrams for Probability Calculations
Brianna is in a lottery for a dorm room in one of two new campus dorms. She will
randomly draw a card to get space in either Xavier or Young (dorms). She will also
draw a card to determine if she gets a room or apartment. The number of cards in
each case is proportional to the number of availabilities. 30% of the available spaces
are in Xavier, 80% of the available spaces in Xavier are apartmentss and 40% of the
spaces in Young are apartments.
It is easy to summarize this information
in a tree diagram.
What is the probability that if Brianna
gets housed in Xavier, she gets a room?
Utilizing Tree Diagrams for Probability Calculations
Brianna is in a lottery for a dorm room in one of two new campus dorms. She will
randomly draw a card to get space in either Xavier or Young (dorms). She will also
draw a card to determine if she gets a room or apartment. The number of cards in
each case is proportional to the number of availabilities. 30% of the available spaces
are in Xavier, 80% of the available spaces in Xavier are apartments and 40% of the
spaces in Young are apartments.
It is easy to summarize this information
in a tree diagram.
What other questions might we ask?
Dependent and Independent Events
Events F and G are independent if P(G | F) = P(G).
If P(G | F) ≠ P(G), then F and G are dependent.
Although this is the formal definition of independent and dependent events, for our
purposes, an intuitive idea of dependence and independence should be good enough.
Intuitively, we say that two events are independent if the occurrence of one of the
events does not affect the probability of the occurrence of the other event.
Dependent and Independent Events
Events F and G are independent if P(G | F) = P(G).
If P(G | F) ≠ P(G), then F and G are dependent.
Although this is the formal definition of independent and dependent events, for our
purposes, an intuitive idea of dependence and independence should be good enough.
Intuitively, we say that two events are independent if the occurrence of one of the
events does not affect the probability of the occurrence of the other event.
Example:
Let A be the event that you get a head when a penny is tossed and B be the event that
you get a tail when a nickel is tossed. Are the events A and B independent?
Dependent and Independent Events
Events F and G are independent if P(G | F) = P(G).
If P(G | F) ≠ P(G), then F and G are dependent.
Intuitively, we say that two events are independent if the occurrence of one of the
events does not affect the probability of the occurrence of the other event.
Example:
Let A be the event that you draw a Spade from a regular deck of cards without
replacement and B be the event that you draw a second Spade from the deck.
Are the events A and B independent?
Dependent and Independent Events
Events F and G are independent if P(G | F) = P(G).
If P(G | F) ≠ P(G), then F and G are dependent.
Intuitively, we say that two events are independent if the occurrence of one of the
events does not affect the probability of the occurrence of the other event.
Example:
Let A be the event that you draw a Spade from a regular deck of cards with
replacement and B be the event that you draw a second Spade from the deck.
Are the events A and B independent?