Lecture 3

Report
Probability and Statistics
Probability
 The likelihood of the occurrence of an event
resulting from statistical experiments
Probability of an event
 It is the sum of the weights of all sample points in
A. Hence
Probability and Statistics
Probability
Example 3.1: A coin is tossed twice. What is the
probability that at least one head occurs?
 If the coin is balanced, each of the outcomes in the
sample space is equally likely to occur. Therefore,
the weight of each outcome is:
Probability and Statistics
Probability
Example 3.1: A coin is tossed twice. What is the
probability that at least one head occurs?
 The event that contains at least one head is:
Probability and Statistics
Probability
Example 3.2: A die is loaded in such a way that an
even number is twice as likely to occur as an odd
number. If E is the event that a number less than 4
occurs on a single toss of the die, find P(E).
 chance of an odd number
 chance of an even number
Probability and Statistics
Probability
Example 3.2: A die is loaded in such a way that an
even number is twice as likely to occur as an odd
number. If E is the event that a number less than 4
occurs on a single toss of the die, find P(E).
Probability and Statistics
Probability
Example 3.2: A die is loaded in such a way that an
even number is twice as likely to occur as an odd
number. If E is the event that a number less than 4
occurs on a single toss of the die, find P(E).
Probability and Statistics
Probability
Example 3.3: from example 3.2 let A be the event that
an even number turns up and let B be the event that a
number divisible by 3 occurs. Find P(AB) and P(AB)
Probability and Statistics
Probability
Example 3.3: from example 3.2 let A be the event that
an even number turns up and let B be the event that a
number divisible by 3 occurs. Find P(AB) and P(AB)
Probability and Statistics
Theorem 2.9
 If an experiment can result in any one of N different
equally likely outcomes, and if exactly n of these
outcomes correspond to event A, then the
probability of event A is
Probability and Statistics
Theorem 2.9
Example 3.4: A mixture of candies contains 6 mints, 4
toffees, and 3 chocolates. If a person makes a random
selection of one of these candies, find the probability
of getting
a) a mint
b) a toffee or a chocolate
Probability and Statistics
Theorem 2.9
Example 3.5: In a poker hand consisting of 5 cards,
find the probability of holding 2 aces and 3 jacks
Probability and Statistics
Theorem 2.9
Example 3.5: In a poker hand consisting of 5 cards,
find the probability of holding 2 aces and 3 jacks
Probability and Statistics
Theorem 2.10
 If A and B are any two events, then
Corollary 1
 If A and B are mutually exclusive, then
Corollary 2
 If A1, A2, A3, … An are mutually exclusive, then
Probability and Statistics
Theorem 2.10
 If A and B are any two events, then
Corollary 3
 If A1, A2, A3, … An is a partition of a sample space S,
then
Probability and Statistics
Theorem 2.11
 For three events A, B, and C,
Probability and Statistics
Theorem 2.11
Example 3.6: The probability that Paula passes mathematics is
2/3, and the probability that she passes English is 4/9. If the
probability of passing both courses is 1/4, what is the probability
that Paula will pass at least one of these courses?
Probability and Statistics
Theorem 2.11
Example 3.7: What is the probability of getting a total of 7 or 11
when a pair of dice are tossed?
 Let A – event that a total of 7 occurs
 Let B – event that a total of 11 occurs
Probability and Statistics
Theorem 2.11
Example 3.7: What is the probability of getting a total of 7 or 11
when a pair of dice are tossed?
 Events A and B are mutually exclusive since a total of 7 or 11
cannot both occur on the same toss
Probability and Statistics
Theorem 2.11
Example 3.8: If the probabilities are, respectively, 0.09, 0.15,
0,21, and 0.23 that a person purchasing a new automobile will
choose the color green, white, red or blue, what is the
probability that a given buyer will purchase a new automobile
that comes in one of those colors?
Probability and Statistics
Theorem 2.12
 If A and A’ are complementary events, then
Example 3.8: If the probabilities that an automobile mechanic
will service 3, 4, 5, 6, 7, 8 or more cars on any given work day are,
0.12, 0.19, 0.28, 0.24, 0.10, and 0.07, what is the probability that
he will service at least 5 cars on his next day at work?
Probability and Statistics
Theorem 2.12
 If A and A’ are complementary events, then
Example 3.8: If the probabilities that an automobile mechanic
will service 3, 4, 5, 6, 7, 8 or more cars on any given work day are,
0.12, 0.19, 0.28, 0.24, 0.10, and 0.07, what is the probability that
he will service at least 5 cars on his next day at work?
Probability and Statistics
Example 3.9: A box contains 500 envelops of which 75 contain
$100 in cash, 150 contain $25, and 275 contain $10. An envelop
may be purchased for $25. Find the probability that the first
envelop purchased contains less than $100.
Probability and Statistics
Example 3.9: A box contains 500 envelops of which 75 contain
$100 in cash, 150 contain $25, and 275 contain $10. An envelop
may be purchased for $25. Find the probability that the first
envelop purchased contains less than $100.
Probability and Statistics
Example 3.10: The probability that an American industry will
locate in Munich is 0.7, the probability that it will locate in
Brussels is 0.4, and the probability that it will locate in either
Munich or Brussels or both is 0.8. What is the probability that
the industry will locate in
a) Both cities?
Given:
b) Neither city?
Probability and Statistics
Example 3.10: The probability that an American industry will
locate in Munich is 0.7, the probability that it will locate in
Brussels is 0.4, and the probability that it will locate in either
Munich or Brussels or both is 0.8. What is the probability that
the industry will locate in
a) Both cities?
Given:
b) Neither city?
Probability and Statistics
Example 3.11: An automobile manufacturer is concerned about a
possible recall of their best-selling four-door sedan. If there were
a recall, there is 0.25 probability that a defect is in a brake
system, 0.18 in the transmission, 0.17 in the fuel system, and
0.40 in some other area.
a) What is the probability that the defect is the brakes or the
fueling system if the probability of defects in both systems
simultaneously is 0.2?
b) What is the probability that there are no defects in either the
brakes or the fueling system?
Probability and Statistics
Example 3.11: An automobile manufacturer is concerned about a
possible recall of their best-selling four-door sedan. If there were
a recall, there is 0.25 probability that a defect is in a brake
system, 0.18 in the transmission, 0.17 in the fuel system, and
0.40 in some other area.
a) What is the probability that the defect is the brakes or the
fueling system if the probability of defects in both systems
simultaneously is 0.15?
Probability and Statistics
Example 3.11: An automobile manufacturer is concerned about a
possible recall of their best-selling four-door sedan. If there were
a recall, there is 0.25 probability that a defect is in a brake
system, 0.18 in the transmission, 0.17 in the fuel system, and
0.40 in some other area.
a) What is the probability that the defect is the brakes or the
fueling system if the probability of defects in both systems
simultaneously is 0.15?
Probability and Statistics
Example 3.11: An automobile manufacturer is concerned about a
possible recall of their best-selling four-door sedan. If there were
a recall, there is 0.25 probability that a defect is in a brake
system, 0.18 in the transmission, 0.17 in the fuel system, and
0.40 in some other area.
b) What is the probability that there are no defects in either the
brakes or the fueling system?
Probability and Statistics
Conditional Probability
 The probability of an event B occurring when it is
known that some event A has occurred
Definition
 The conditional probability of B, given A, denoted
by P(B|A), is defined by
if P(A) > 0
Probability and Statistics
Example 3.12: Suppose that our sample space S is the population
of adults in a small town who have completed the requirements
for a college degree. We shall categorize them according to sex
and employment status:
Male
Female
Total
Employed Unemployed
460
40
140
260
600
300
Total
500
400
900
One of these individuals is to be selected at random for a tour
throughout the country to publicize the advantages of
establishing new industries in the town.
Probability and Statistics
Example 3.12: Suppose that our sample space S is the population
of adults in a small town who have completed the requirements
for a college degree. We shall categorize them according to sex
and employment status:
Male
Female
Total
Employed Unemployed
460
40
140
260
600
300
Total
500
400
900
P (M) = probability that a man is chosen
P (E) = probability that the one chosen is employed
Probability and Statistics
Example 3.12: Suppose that our sample space S is the population
of adults in a small town who have completed the requirements
for a college degree. We shall categorize them according to sex
and employment status:
Male
Female
Total
Employed Unemployed
460
40
140
260
600
300
Total
500
400
900
Probability and Statistics
Example 3.13: The probability that a regularly scheduled flight
departs on time is P(D) = 0.83; the probability that it arrives on
time is P(A) = 0.82; and the probability that it departs and arrives
on time is P(D  A) = 0.78. Find the probability that a plane
a) Arrives on time given that it departed on time
b) Departs on time given that it arrived on time
c) Arrives on time given that it departed late
Probability and Statistics
Example 3.13: The probability that a regularly scheduled flight
departs on time is P(D) = 0.83; the probability that it arrives on
time is P(A) = 0.82; and the probability that it departs and arrives
on time is P(D  A) = 0.78. Find the probability that a plane
a) Arrives on time given that it departed on time
b) Departs on time given that it arrived on time
c) Arrived on time given that it departed late
Probability and Statistics
Example 3.13: The probability that a regularly scheduled flight
departs on time is P(D) = 0.83; the probability that it arrives on
time is P(A) = 0.82; and the probability that it departs and arrives
on time is P(D  A) = 0.78. Find the probability that a plane
a) Arrives on time given that it departed on time
b) Departs on time given that it arrived on time
c) Arrived on time given that it departed late
Probability and Statistics
Example 3.13: The probability that a regularly scheduled flight
departs on time is P(D) = 0.83; the probability that it arrives on
time is P(A) = 0.82; and the probability that it departs and arrives
on time is P(D  A) = 0.78. Find the probability that a plane
a) Arrives on time given that it departed on time
b) Departs on time given that it arrived on time
c) Arrived on time given that it departed late
Probability and Statistics
Example 3.14: from example 3.2 - A die is loaded in such a way
that an even number is twice as likely to occur as an odd number.
Let B be the event of getting a perfect square, and A be the event
of getting a number greater than 3. Find the probability of
getting a perfect square given that the number is greater than 3.
Probability and Statistics
Example 3.14: from example 3.2 - A die is loaded in such a way
that an even number is twice as likely to occur as an odd number.
Let B be the event of getting a perfect square, and A be the event
of getting a number greater than 3. Find the probability of
getting a perfect square given that the number is greater than 3.
Probability and Statistics
Independent Event
 The probability of an event B occurring when it is
known that some event A has occurred
Example 3.15: 2 cards are drawn in succession from an ordinary
deck with replacement.
 Let A – the first card is an ace
 Let B – the second card is a spade
Probability and Statistics
Independent Event
 The probability of an event B occurring when it is
known that some event A has occurred
Example 3.15: 2 cards are drawn in succession from an ordinary
deck with replacement.
 Let A – the first card is an ace
 Let B – the second card is a spade
Probability and Statistics
Multiplicative Rules
Theorem 2.13
 If in an experiment the events A and B can both
occur, then
Probability and Statistics
Theorem 2.13
Example 3.16: A fuse box contains 20 fuses, of which 5 are
defective. If 2 fuses are selected at random and removed from
the box in succession without replacing the first, what is the
probability that both fuses are defective?
 Let A – the first fuse is defective
 Let B – the second fuse is defective
Probability and Statistics
Theorem 2.13
Example 3.17: One bag contains 4 white balls and 3 black balls,
and a second bag contains 3 white balls and 5 black balls. One
ball is drawn from the first bag and placed unseen in the second
bag. What is the probability that a ball now drawn from the
second bag is black?
Probability and Statistics
Theorem 2.13
Example 3.17: One bag contains 4 white balls and 3 black balls,
and a second bag contains 3 white balls and 5 black balls. One
ball is drawn from the first bag and placed unseen in the second
bag. What is the probability that a ball now drawn from the
second bag is black?
Probability and Statistics
Theorem 2.14
 Two events A and B are independent if and only if
Example 3.18: A small town has one fire engine and one
ambulance available for emergencies. The probability that a fire
engine is available when needed is 0.98, and the probability that
the ambulance is available when called is 0.92. In an event of an
injury resulting from a burning building, find the probability that
both the ambulance and the fire engine will be available.
Probability and Statistics
Theorem 2.15
 If in an experiment, the events A1, A2, A3,…Ak can
occur, then
 If the events A1, A2, A3,…Ak are independent, then
Probability and Statistics
Theorem 2.15
Example 3.19: Three cards are drawn in succession, without
replacement, from an ordinary deck of playing cards. Find the
probability that the event A1  A2  A3 occurs, where A1 is the
event that the first card is a red ace, A2 is the event that the
second card is a 10 or jack, and A3 is the event that the third card
is greater than 3 but less than 7.
Probability and Statistics
Theorem 2.15
Example 3.19: Three cards are drawn in succession, without
replacement, from an ordinary deck of playing cards. Find the
probability that the event A1  A2  A3 occurs, where A1 is the
event that the first card is a red ace, A2 is the event that the
second card is a 10 or jack, and A3 is the event that the third card
is greater than 3 but less than 7.
Probability and Statistics
Theorem 2.16: Total Probability
 If the events B1, B2, …, Bk constitute a partition of the sample S
such that P(Bi)  0 for i = 1,2,…k, then for any event A of S,
Probability and Statistics
Theorem 2.16: Total Probability
Example 3.20: In a certain plant, three machines, B1, B2, and B3,
make 30%, 45%, and 25%, respectively, of the products. It is
known from past experience that 2%, 3%, and 2% of the products
made by each machine, respectively, are defective. Now,
suppose that a finished product is randomly selected. What is
the probability that it is defective?




A – the product is defective
B1 – the product is made by machine B1
B2 – the product is made by machine B2
B3 – the product is made by machine B3
Probability and Statistics
Theorem 2.16: Total Probability
Example 3.20: In a certain plant, three machines, B1, B2, and B3,
make 30%, 45%, and 25%, respectively, of the products. It is
known from past experience that 2%, 3%, and 2% of the products
made by each machine, respectively, are defective. Now,
suppose that a finished product is randomly selected. What is
the probability that it is defective?
Probability and Statistics
Theorem 2.17: Baye’s Rule
 If the events B1, B2, …, Bk constitute a partition of the sample S
such that P(Bi)  0 for i = 1,2,…k, then for any event A of S such
that P(A) )  0 ,
Probability and Statistics
Theorem 2.17: Baye’s Rule
Example 3.21: from example 3.20 - if a product were chosen at
random and found to be defective, what is the probability that it
was made by machine B3?
Probability and Statistics
Theorem 2.16: Total Probability
Example 3.22: Police plan to enforce speed limits by using radar
traps at 4 different locations within the city limits. The radar
traps at each locations L1, L2 , L3 and L4 are operated 40%, 30%,
20%, and 30% of the time, and if a person who is speeding on his
way to work has probabilities of 0.2, 0.1, 0.5 and 0.2,
respectively, of passing through these locations,
a. What is the probability that he will receive a speeding ticket?
b. If the person received a speeding ticket on his way to work,
what is the probability that he passed through the radar trap
located at L2?

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