Download Conditional Probability

CONDITIONAL
PROBABILITY
Conditional Probability
• Knowledge that one event has
occurred changes the likelihood that
another event will occur.
• Denoted: P(A|B)
• The probability of A given than B has
already occurred.
Conditional Probability
P A  B 
P A / B  
P B 
Ex: Population has .1% of all individuals having a
certain disease. A test is available and 80% of those
who test positive actually have the disease.
E = the individual has the disease
F = the individual’s diagnostic test is positive
P(E) =
P(E | F) =
Before the test, the occurrence of E was
highly unlikely, after the test turns out positive
it’s highly likely.
Example: Titanic
1st Class
2nd
Class
3rd
Class
Survived
203
118
178
499
Died
122
167
528
817
325
285
706
1316
P(3rd class)
P(3rd class | survived)
P(1st class | survived)
P(died | 2nd class)
Students at the University of New Harmony received 10,000 course grades
last semester
Total
6300
1600
2100
Total 3392 2952
3656
10000
• Find P(lower than B)
• Find P(E|Low) and P (Low|E).
• Which of the above tells you whether this college’s
engineering students tend to earn lower grades than
students in liberal arts.
Find the probability of drawing a 3 of diamonds if you already
know that it’s a red card.
Find probability of a family having 2 girls given that they have
at least 1 girl.
A consumer organization estimates that over a 1-year period 17%
of cars will need to be repaired once, 7% will need repairs twice,
and 4% will require 3 or more repairs.
• Probability that a car chosen at random will need
a. No repairs
b. No more than one repair
c. Some repairs
You randomly draw a card at random from a standard deck
of 52 cards.
• P(heart | red)
• P(red | heart)
• P(ace | red)
• P(queen | face card)
70% of kids who visit a doctor have a fever, and 30% of kids with
a fever have sore throats. What’s the probability that a kid who
goes to the doctor has a fever and a sore throat?
In a class of 45 students 18 like apples and 32 like bananas
and 5 dislike both fruits. If a students is randomly
selected, find the probability that the student:
• likes both fruit
• Likes at least one fruit
• Likes Bananas given that
they like apples
• Dislikes apples given that
they like bananas.
Independent Events
• Two events A and B are independent if the
occurrence of one event has no effect of the
chance that the other event will happen.
• In other words, events A and B are
independent if
P A / B  P A and PB / A  PB
Using the blocks:
• What’s the probability of drawing a green?
• If I replace the block and draw again what is the
probability that the second block is green?
Using the blocks:
• What’s the probability of drawing a green?
• If I do not replace the block and draw again what is the
probability that the second block is green?
So….
• Sampling with replacement is ____________
• Sampling without replacement is
________________
If I roll 2 dice, what’s the probability that I get a
three on both? Independent or Dependent?
Is there a relationship between gender and handedness? To find
out, we used CensusAtSchool’s Random Data Selector to choose
a SRS of 50 Australian high school students who completed a
survey.
Dominant hand
Gender
Right
Left
Total
Male
20
3
23
Female
23
4
27
Total
43
7
50
If two events are not independent, does that mean
than there actually is a relationship between the
two variables?
• No…not necessarily
Independent or not independent?
• Shuffle a standard deck of cards, and turn
over the top card. Put it back in the deck,
shuffle again, and turn over the top card.
Define events
• A: first card is a heart
• B: second card is a heart
Independent or not independent?
• Shuffle a standard deck of cards, and turn
over the top two cards, one at a time.
Define the events
• A: first card is a heart
• B: second card is a heart
Independent or not independent?
• The 28 students in a class completed a brief
survey. One of the questions asked whether each
student was right- or left-handed. Choose a
student from the class at random. The events of
interest are “female” and “right-handed.”
Gender
Handedness Female
Left
Right
3
18
Male
1
6
Multiplication Rule
• Dependent:
• Independent:
P(King and then King) with
replacement.
P(king & King) without replacement.
P(king and 4) without replacement
P(E)=.2
P(F)=.3
• Find P(E and F) if they are independent.
If P(A) = 0.2 and P(B) = .4 and P(A  B) = p. Find p
if
• A and B are mutually exclusive
• A and B are independent
Using a tree diagram, what’s the probability of
getting two heads when you toss a fair coin?
Suppose that 60% of all customers of a large insurance agency
have automobile policies with the agency, 40% have
homeowner’s policies, and 25% have both types of policies. If a
customer is randomly selected, what is the probability that he or
she has at least one of these two types of policies with the
agency? (Look at Venn Diagram)
The probability of purchasing a Sony DVD player is
0.70. The probability of an extended warranty being
purchased when a Sony DVD player is bought is 0.20.
Find the probability that a person buys a Sony DVD
and the extended warranty.
Homework
• Page 329 (63, 65, 67, 69, 71, 73, 75, 79, 106)