Conditional Probability
Let A and B be any two nonempty events.
We write P(A | B ) or P( A / B ) to mean “the probability of A given that B has occurred.”
P( A Ç B) means “the probability that both A and B occur”
P( A È B) means “ the probability that either A , B, or both occur”
While every group of disjoint sets happen to be _____________________________
not every pair of sets that intersect are _______________
but every pair of sets that are _______________________ will have an intersection.
We can write each of the following probabilities regardless of the relationship between A and B.
1) P ( A È B ) = P(A) + P(B) – P(A Ç B )
2) P ( A / ) = 1 – P( A ) , and
*** 3) P( A Ç B ) = P( A ) · P( A | B ),
Def. We define P(A | B ) as
P( A Ç B)
P(A | B ) = ---------------
P( B )
ex. If P( A) = 0.4, and P( B) = 0.8, can A and B be mutually exclusive ? Why or Why not ? _____
Using the probabilities above assume that P( A È B ) = 0.9. Find
P( A | B ) = ______________ what about P( B | A ) = _____________
ex. If P( A ) = 0.6 , P( B ) = 0.5 , and P( A Ç B ) = 0.3, then find
a) P( A È B ) = _______________ b) P ( A | B ) = _______________
c) P( B | A ) = _______________
ex. Suppose that A and B were mutually exclusive events with neither being nonempty.
Find P( A | B ) = __________
ex. Suppose that A and B were independent events. find
a) P( A | B ) = _______________ b) P( A Ç B ) = _________________
This is how you determine if two events are independent or not .
____________________________________
What is the connection between independent and mutually exclusive ? Any ? relationship ?
ex. Four lights are independent of each other. What is the probability that a person will
( Assume that the probability that a light is green is 3/5 )
a) get all four green lights
b) get the first light to be green
It is useful to use a tree diagram to answer questions about conditional probability.
The following example illustrates what each part of the branches represents.
ex. Two balls are drawn from an urn that contains 4 red, 7 blue, and 9 white balls. If the second ball is drawn without
replacement find the probability that
both balls are red ? ___________ the first is red and the second one is white ? _____________
the second one is white if you know that the first one is red ? _________________
the second one is white ? _______________ the first one is red if you know that the second one was white ? ____
Factorials
Recall:
n! = n(n-1)(n-2)·... · (2)(1) , also we define 0! = 1! = 1
ex. 6 ! = ___________ 8 ! / 6 ! = _____________ 4 ( 3 ! + 2 ! ) = __________
Defined. 0 ! = ___________ 1 ! = ____________
Fundamental Counting Principle: Decisions D1, D2, ... can be done together in r1·r2·r3· ....
good examples: how many different menus are possible if there are four salads, three meats , and six desserts
and the diner must eat only one of each type ?
___________
how many different codes are possible if the code consists of sex, two digit age, and two digits soc.
sec. # ?
___________
Permutations: no repetitions, order matters, ABDC is different than ACBD
good examples: how many ways can a Pres., Vice P be chosen from a group of 12 _________
how many ways can 6 books be arranged on a shelf ? __________
how many different “words” can be made using the blocks ABCD ? ________
how many using only two of the blocks ? _______
Notation:
Formula:
Combinations: no repetitions, order does not matter ABDC is the same as ACBD
good examples: Two new federal judges will be appointed from a group of 20 state judges. How many different selections
are possible ? If 12 are men and 8 are women , what is the probability that they are both women ?
_________
How many ways can a committee of 4 be appointed from a group of 8 men and 4 women ? ________
What is the probability that all four are men ?
What is the probability that exactly two are women ? __________
What is the probability that at least one is a man ? ____________
Consider all five card draws from a standard deck of cards. How many different hands are possible ?
What is the probability that all are hearts ? ______________
What is the probability that at least one is a heart ? ____________
What is the probability that exactly three are hearts ? __________
Permutations:
1) no repetitions and 2) order matters
ex. { a, b, c , d } è some three letter permutations è
ex. How many ways can a Chairman and a Vice Chairman of the Board be selected if they are to come from a
group of a 10-member board ?
Formula: P( n, r ) = n P r =
Combinations:
1) no repetitions and 2) order does not matter
ex. { a, b, c, d } è some three letter combinations è
ex. How many ways can a two member committee be chosen from a 10-member board ?
Formula: C(n,r ) = _________________ = ____________________ =
Other examples:
1) 34/ 546 : Eight horses are entered in a race. In how many ways can the horses finish ?
What is the probability that the top three favored horses finish in the order 1,2,3 ?
What about the probability that the top three horses finish in the top three ?
2) 41/ 546 How many ways can a 10 –question multiple be answered if each question has four different answers ?
What is the probability that the first and last questions are correct ?
3) 48/ 547 A poker hand consists of 5 cards dealt from a deck of 52 cards. How many different poker hands are possible ?
How many with four aces ?
What is the probability that you have four aces ? ____________
4) In how many ways can a hand consisting of 6 spades, 4 hearts, 2 clubs, and 1 diamond be selected from a deck of 52
cards ?
5) How many different sequences are possible in a 4 rolls of a fair six-sided die ? _____________________
What is the probability that the sequence consists of all sixes ? ____________
What is the probability that the sequence consists of no sixes ? _______________
4/551: Two men and a woman are lined up to have their picture taken. If they are arranged at random, what is the probability
that
a) the woman will be on the left in the picture ? b) the woman will be in the middle of the picture ?
8/551 Keys for older General Motors cars had six parts, with three patterns each.
a) How many different key designs are possible for these cars ?
b) If you find an older GM key and you own an older GM car, what is the probability that it will fit your trunk ?
14/551 10 –question with 15 possible matches, no repetitions – what is the probability of guessing and getting every answer
correct ?
20/552 12 girls at random from a freshman class : 200 freshman girls, including 20 from minorities, and the principal would
like at least one minority girl to have this honor. If he selects the girls at random, what is the probability that
a) he will select exactly one minority girl ?
b) he will select no minority girls ?
c) he will select at least one minority girl?
32/553
Birthday Problem:
What is the probability that
Two people will have the same birthday. è___________
Two people will not have the same birthday è________________
At least two people will have the same birthday in a group of 10 ? è _____________
HW: page 551 1, 3, 7, 9, 11, 13, 17, 19, 23, 27, 31, 33, 35 page 546: 21,23,25, 32, 37, 40, 41, 45, 46, 49, 53
Day 12 – June 12, 2002
Markov – Chains –
Transition Matrix: has probabilities (transition probabilities)
they describe the probability in moving from one state to another.
Credit Card: Let C = event credit card is used. Previous experience tells a company that
P( C next month | C prev. month ) = 0.8 P( C next month | C/ prev. month ) = 0.3
P( C/ next month | C prev. month ) = 0.2 P( C/ next month | C / prev. month ) = 0.7
This can be illustrated with a tree diagram as well as with a matrix.
Transition next month
matrix uses card card is not used
Given card is used 0.8
Month card in not used 0.3
Initial –probability vector
Prob. woman uses card = 0.9 prob. woman does not use card: _____
What are the probabilities for the second month ? ____________
Look at example on page 558: [ 0.6 0.3 0. 1 ]
0.2 0.6 0.2
0.1 0.5 0.4
0.1 0.1 0.8
What are the probabilities after one generation ?
Steady State Vectors ---
HW: page 560 1, 4, 5, 8, 9, 23,
page 553 9, 12, 25, 27, 29, 30, 31, 33, 35,
Bernoulli Experiment:
When we look at an experiment we can usually separate into two parts – two types of outcomes.
We will define the two outcomes as ; a success (s) and a failure (f)
A Bernoulli experiment has two outcomes labeled
s for ______________ and f for _____________
We let P(s) = p and P(f) = q. è property: p + q = _________
ex. A single- sided- loaded die is rolled . ( outcome is a four or not ) ex. A fair coin is tossed ( a head or not )
ex. A card is chosen from a standard deck ( an ace or not )
ex. A patient checks into the emergency room – and is observed for serious injuries. ( serious or not )
ex. A person is driving down a street that has five traffic lights. A light is green three minutes and red for one
minute. What is the probability that the person will
a) get only one red light ? b) at least one red light ?
ex. Six different races are being held at the race track (horses). In each race there are ten horses and each
horse is equally likely to win (random). What is the probability that a person picking the horse at random
will pick
a) all winners ? _______ b) exactly four winning horses ? _________
Let p be the success probability and q be failure probability of a Bernoulli experiment
If we repeat the Bernoulli experiment n times – the resulting experiment is called a
_____________________
Notice: the n Bernoulli trials must be independent of each other ( coin tosses, rolls of a die, … )
A binomial experiment is a sequence of independent Bernoulli trials. If we let x represent the number of success in n Bernoulli trials, then
0 £ x £ n , where n is a whole number
We can find the probability of x success by the following formula; C( n, x ) px qn- x
ex. A coin is tossed five times. What is the probability of getting at least four heads ?
ex. Consider 3-child families. What is the probability that exactly 2 are male children ?
ex. A die is rolled ten times. What is the probability that you will roll four sixes ?
ex. A card is selected at random from a standard deck of cards that consists only of face cards and aces. The
card is then replaced. What is the probability that in 10 draws
a) you will draw six aces ? _________ b) you will draw six face cards ? ______
ex. Last year’s records show that out of 500 children examined by a doctor – 12 were not up to date on
their vaccination records. If ten of these children are chosen at random , what is the probability
that two of them will have needed shots.
ex. A loaded coin with P(heads) = 2/5 is tossed four times. Find the following probabilities.
a) no tails _____________________ b) exactly 2 heads _________________
c) at least one head __________________
ex. A six-sided die has the following probability model; P ( x ) = x / 21 , where x represents its face 1, 2,3, 4, 5, 6
The die is rolled five times. What is the probability that
a) none of the rolls land in a six ? ______________ b) exactly four of the rolls are sixes ? ___________
c) at least one of the rolls is a six ? _____________
ex. A standard deck of cards is shuffled and a card is chosen at random, recorded and replaced.
Suppose four cards are selected in this manner . What is the probability that
a) all are aces ? b) all are diamonds ?
c) at least one is red ?
Day 13 – June 13, 2002
Def. A ____________ _____________ is a rule that assigns a numerical value to each outcome of an experiment.
Now every sample space that we look at is written or can be written in terms of numerical values.
ex.1 A loaded coin is tossed three times and the sequence is recorded. What is a good sample space and how could we
assign a numerical value to it ?
ex2. A pair of dice is rolled four separate times. Suppose you are interested in seeing a double. Describe a sample space
to which each outcome can be assigned a numerical value.
ex3 . A card is picked at random from a well-shuffled deck of cards. The card’s value is recorded and replaced. The deck
is shuffled and again. This pattern continues until an ace comes up. Describe the sample space so that a numerical
value can be assigned.
ex4. A class consists of 45 students. Describe a sample space that tells the number of days a student has been absent
during the lass 18 days of class.
ex5. Record the exact amount of time that one of the forty-five students can hold their breath during the 50 minute class.
Describe this experiment in such a way that you can assign a value to all the outcomes.
We have seen three different kinds of ______________ ___________________.