Transcript Conditional Probability 3.2

CONDITIONAL PROBABILITY
3.2
Raise your hand when you are
finished reading.
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 by P(B/A)
Conditional Probability
STARTING SIMPLE
There are more red coins among the large coins
than among the small coins. When randomly selecting
a coin from the box, the probability of obtaining a red
coin is what?
P(red) = 5/10 = 1/2
because there are 5 red coins in a total of 10 coins.
NOW ASSUME THAT A COIN IS PICKED BY A
BLINDFOLDED PERSON.
The coin can be felt to be large, but the color
is unknown.
What is the probability that the coin is red and
large?
IN THIS CASE P(RED AND LARGE) = 3/5
Because all the small coins are ruled out
and only the large coins are considered.
There are 5 large coins, 3 of which happen
to be red.
DEPENDENT
The probability of red changed when it was known
that the coin was large.
P(red and large) ≠ P(red)
This is an indication that color and size are NOT
independent.
They are DEPENDENT
If P(B/A)≠P(B) or when P(A/B)≠P(A)
INDEPENDENT
Two events are independent when the
occurrence of one of the events does not
affect the probability of the occurrence of
the other event.
Two events A and B are independent when
P(B/A)=P(B)
or when P(A/B)=P(A)
INDEPENDENT
P(B/A)=P(B) or when P(A/B)=P(A)
“the probability of B, given A”
A is the event that happened first
B is the event that is happening second
Because they are independent
The first event does not affect the
Probability of the second event.
ROLL THE DICE
Roll the first Dice then roll the second dice
1.) What is the probability that you will roll the
same number again?
2.) What is the probability that you will roll an
even number?
DEPENDENT OR INDEPENDENT
1.) Returning a movie after the due date and
receiving a late fee.
2.) A father having hazel eyes and a daughter having
hazel eyes.
3.) Select a 10 from a deck of cards, replacing it, and
then selecting a 10 from the deck again.
THE MULTIPLICATION RULE
The probability that two events A and B will occur
in sequence is
P(A and B) = P(A)∙P(B/A)
If events A and B are independent, then the rule can
be simplified to
P(A and B) = P(A)∙P(B)
USING THE MULTIPLICATION RULE
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.
P(K and Q) = P(K)∙P(Q/K)
=
𝟒
𝟓𝟐
∙
𝟒
𝟓𝟏
=
𝟏𝟔
𝟐𝟔𝟓𝟐
= 𝟎. 𝟎𝟎𝟔
USING MULTIPLICATION RULE
To Find Probabilities
For anterior cruciate ligament (ACL)
reconstructive surgery, the probability that
the surgery is successful is 0.95.
Find the probability that three ACL surgeries
are successful.
EXAMPLE 1
The probability that each ACL surgery is successful is
0.95. The chance of success for one surgery is
independent of the chances for the other surgeries.
P(three surgeries are successful)=(0.95)(0.95)(0.95)
≈ 0.857
The probability that all three surgeries are successful is
about 0.857.
EXAMPLE 2
To Find Probabilities
For anterior cruciate ligament (ACL) reconstructive
surgery, the probability that the surgery is successful is
0.95.
2.) Find the probability that none of the three ACL
surgeries are successful.
EXAMPLE 2 CONTINUED
Because the probability of success for one surgery is
0.95, the probability of failure for one surgery is
1 − 0.95 = 0.05
P(none of the three are successful)
= (0.05)(0.05)(0.05) ≈ 0.0001
EXAMPLE 3
For anterior cruciate ligament (ACL) reconstructive
surgery, the probability that the surgery is successful is
0.95.
3.) Find the probability that at least one of the three
ACL surgeries is successful.
EXAMPLE 3 CONTINUED
The phrase “at least one” means one or more. The
complement to the event “at least one is successful”
is the event “none are successful.” use the
complement to find the probability.
P(at least one is successful) = 1 −
𝑃(𝑛𝑜𝑛𝑒 𝑎𝑟𝑒 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙)
≈ 1 − 0.0001
≈ 0.9999
USING THE MULTIPLICATION RULE
About 16,500 U.S. medical school seniors applied to
residency programs in 2012. Ninety-five percent of the
seniors were matched wit residency positions. Of those,
81.6% were matched with one of their top three
choices. Medical students rank the residency programs
in their order of preference, and program directors in
the U.S. rank the students. The term “match” refers to
the process whereby a student’s preference list and a
program director’s preference list overlap, resulting in
the placement of the student in a residency position.
𝑃 𝑅 = .95
𝑃(T/𝑅) = .816
QUESTION 1
Find the probability that a randomly selected senior
was matched with a residency position and it was
one of the senior’s top three choices.
𝑃 𝑆 𝑎𝑛𝑑 𝑇 = 𝑃 𝑆 ∙ 𝑃 T/R = 0.95 0.816
= 0.775
QUESTION 2
Find the probability that a randomly selected senior
who was matched with a residency position did not
get matched with one of the senior’s top three
choices.
1 − 𝑃(T/R)= 1 − 0.816 = 0.184
QUESTION 3
Would it be unusual for a randomly selected senior to
be matched with a residency position and that it was
one of the senior’s top three choices?
Similar to Question 1.
It is not unusual because the probability of a
senior being matched with a residency position that
was one of the senior’s top three choices is about
0.775≈ 77.5%. 77.5% is not usual. The unusual %’s are
3% or less and 97% or more.
TRY
http://www.regentsprep.org/regents/math/algebra/
APR3/PracCond.htm
#’s 1 and 2