Download Section 3-2 Notes Outline

Probability and Statistics
Chapter 3 Notes
Section 3-2
I. Conditional Probability
A conditional probability is the probability of an event occurring, given that another event has already occurred.
The conditional probability of event B occurring, given that event A has occurred, is denoted 𝑃(𝐵|𝐴) and is read as
“probability of B, given A.”
II. Independent and Dependent Events
Two events are
if the occurrence of one of the events
affect the probability
of the occurrence of the other event.
Two events A and B are
if 𝑃(𝐵|𝐴) = 𝑃(𝐵), or if 𝑃(𝐴|𝐵) = 𝑃(𝐴).
In other words, event B is
likely to occur whether event A has
or not.
Events that are not independent are
.
To determine if A and B are independent, first calculate
, the probability of event
. Then calculate
, the probability of
, given
.
If the two probabilities are
, the events are
.
If the two probabilities are not
, the events are
.
III. The Multiplication Rule
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴)
If Events A and B are dependent;
1) Find the probability the first event occurs
2) Find the probability the second event occurs given the first event has occurred.
3) Multiply these two probabilities to find the probability that both events will occur in sequence.
If Events A and B are independent, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵).
This simplified rule can be extended for any number of independent events, just like the Fundamental Counting
Principle could be extended.
Example 1 (Page 149)
1) Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given
that the first card is a king. Assume that the king is not replaced (it’s not put back into the deck before the queen is
drawn.
2) The table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in
the child. Find the probability that a child has a high IQ given that the child has the gene.
High IQ
Normal IQ
Total
Gene Present
33
39
72
Gene Not Present
19
11
30
Total
52
50
102
SOLUTIONS:
1) The probability of drawing a queen after a king (or any other card) has been taken out of the deck is
, or about
.
2) There are
children who have the gene. They are our
. Of these,
have a high IQ.
So, 𝑃(𝐵|𝐴) =
≈
.
Example 2 (Page 150)
Decide whether the events are independent or dependent.
1) Selecting king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B).
2) Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B).
SOLUTIONS:
1) The probability of pulling a queen out of the deck is
.
The probability of pulling a queen out of the deck after a king has already been removed is
.
Since these probabilities are
, the events are
.
2) The probability of obtaining a 6 is
.
The probability of obtaining a 6 given that the coin came up heads is
.
Since these probabilities are
, the events are
.
Example 3 (Page 151)
1) Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king
and then selecting a queen.
2) A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.
SOLUTIONS:
1) Because the first card is not replaced, the events are
.
𝑃(𝐾 𝑎𝑛𝑑 𝑄) =
*
*
=
≈
2) The two events are
.
𝑃(𝐻 𝑎𝑛𝑑 6) = 𝑃(𝐻) ∗ 𝑃(6)
*
=
≈
Example 4 (Page 152)
1) A coin is tossed and a die is rolled. Find the probability of getting a tail and then rolling a 2.
2) The probability that a particular knee surgery is successful is 0.85. Find the probability that three surgeries in a row are
all successful.
3) Find the probability that none of the three knee surgeries is successful.
4) Find the probability that at least one of the three knee surgeries is successful.
SOLUTIONS
1) The probability of getting a tail and then rolling a 2 is
𝑃(𝑇 𝑎𝑛𝑑 2) = 𝑃(𝑇) ∗ 𝑃(2)
*
=
≈
2) The probability that each knee surgery is successful is 0.85. Since these events are
, the
probability that all three are successful is found by
their probabilities together.
≈
The probability that all three surgeries are successful is about
.
3) Because the probability of success for one surgery is .85, the probability of failure for one surgery is
.
This is true because failure is the
of success.
The probability that each knee surgery is not successful is
. Since these events are
,
the probability that all three are not successful is found by
their probabilities together.
≈
The probability that none of the surgeries are successful is about
.
4) The phrase “at least one” means
. The complement to the event “at least one is successful”
is the event “
”.
Using the complement rule, we can simply
the probability that none were successful from
to find the probability that at least one was successful.
=
.
The probability that at least one of the surgeries is successful is about
.
Example 5 (Page 153)
More than 15,000 medical school seniors applied to residency programs in 2007. Of those, 93% were matched to a
residency position. 74% percent of the seniors matched to a residency position were matched to one of their top two
choices.
1) Find the probability that a randomly selected senior was matched to a residency program and it was one of the
senior’s top two choices.
2) Find the probability that a randomly selected senior that was matched to a residency program did not get matched
with one of the senior’s top two choices.
3) Would it be unusual for a randomly selected senior to result in a senior that was matched to a residency position
and it was one of the senior’s top two choices?
SOLUTION:
1) These two events are
:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴) =
*
≈
.
2) To find this probability, use the complement:
𝑃(𝐵′ |𝐴) = 1 − 𝑃(𝐵|𝐴) = 1 =
.
3)
:
This event occurs around
% of the time.