midterm part 3 new

1. Two balls are drawn successively without replacement from a box

which contains 4 white balls and 3 red balls. Find the probability

that

(a) the first ball drawn is white and the second is red;

(b) both balls are red.

(a) The second event is dependent on the first.P(E1) = P(white) = 4/7

There are 6 balls left and out of those 6, three of them are red. So the probability that the second one is red is given by: P(E2 | E1) = P(red) = 3/6 = 1/2

Dependent events, so

P(E1 and E2) = P(E1) × P(E2 | E1) = 4/7 × 1/2 = 2/7

Rule of Multiplication If events A and B come from the same sample space,

the probability that both A and B occur is equal to the probability the event A

occurs times the probability that B occurs, given that A has occurred.

P(A ∩ B) = P(A) * P(B|A)

A. A jar contains 6 red marbles and 4 black marbles. Two marbles are drawn

without replacement from the urn. What is the probability that both of the

marbles are black?

Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:

P(A) = 4/10.

After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9. Therefore, based on the rule of multiplication: P(A ∩ B) = P(A) P(B|A)

P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15

Suppose we repeat the experiment of Example A; but this time we select marbles

with replacement. That is, we select one marble, note its color, and then replace it

in the urn before making the second selection. When we select with replacement,

what is the probability that both of the marbles are black?

there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.

After the first selection, we replace the selected marble; so there are still 10

marbles in the urn, 4 of which are black. Therefore, P(B|A) = 4/10.

Therefore, based on the rule of multiplication:

P(A ∩ B) = P(A) P(B|A)

P(A ∩ B) = (4/10)*(4/10) = 16/100 = 4/25

Independent and Dependent events

DEPENDENT Events

Two events are said to be dependent if the occurrence of one affects

the occurrence of the other.

A bag contains 5 white marbles, 3 black marbles and 2 green marbles. In

each draw, a marble is drawn from the bag and not replaced. In three draws,

find the probability of obtaining white, black and green in that order.

We have 3 dependent events.

=5/10 x 3/9 x 2/8 = 1 /24

Independent Events

Events are said to be independent if the occurrence of one event does

not affect the occurrence of others.

1. A fair die s tossed twice. Find the probability of getting a 4,5 or 6 on

the first toss and a 1,2,3 or 4 on the second toss

= 3/6 x 4/6 = 12/36 or 1/3

If A and B are independent events, then probability of both occurring is: P(A) x (B)

2. What is the probability of rolling a 3 with a dice, and drawing a 3

from a deck of cards?

P(A n B) = 1/6 x 4/52 = 1/78

1.A bag contains 5 white and 8 black balls, 2 balls are drawn at

random. Find

a)The probability of getting both the balls white, when the first ball

drawn, is replaced.

b)The probability of getting both the balls white, when the first ball is

not replaced.

2.Three cards are drawn from an ordinary deck and not replaced. Find

the probability of these.

a. Getting 3 jacks

b. Getting an ace, a king, and a queen in order

c. Getting a club, a spade, and a heart in order

d. Getting 3 clubs

3. Suppose that we have a fuse box containing 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?

4. A coin is flipped and a die is rolled. Find the probability of getting a

head on the coin and a 4 on the die.

5. The small town has one fire engine and one ambulance available

for emergencies, the probability that the fire engine is available

when needed is 0.98, and the probability that the ambulance is

available when called is 0.92. In the event of an injury resulting

from a building burning, find the probability that both the ambulance

and fire engine will be available.

6. 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 place unseen

in the second bag. What is the probability that a ball

drawn now from the second bag is black

7. Three cards are drawn in succession, without

replacement, from an ordinary cards. Find the

probability that A ∩ B ∩ C occurs where A is the event

that first card is a red ace, event B the second card is

10 or a jack, and events C for third card is greater than

3 but less than 7.

CALCULATING PROBABILITIES FOR COMBINATIONS OF EVENTS

A. CONDITIONAL PROBABILITIES

Given two events A and B, if we want to determine the

probability of the intersection of the two events, P(A∩B)

we answer this question: What is the probability that

events A and B will occur? If on the other hand, we want to

determine a conditional probability for these events, we

answer a related but different question: What is the

probability of A occurring given that B is known to have

occurred? Or the reverse question: What is the probability

of B given that A is known to have occurred?

The conditional probability of A, given B, denoted by :

P(AB) P(AB) = ----------------------- provided that P(B) 0

P(B)

P(AB) P(BA) = ----------------------- provided that P(A) 0

P(A)

The conditional probability of B, given A, denoted by :

1. The probability that a regularly scheduled flight departs on time

is 0.83, the probability that it arrives on time is 0.92, and the

probability that it departs and arrives on time is 0.78. Find the

probability that a plane (a) arrives on time given that it departed

on time, and (b) departed on time given that it has arrived on

time

2. The probability that an automobile being filled with gasoline will

also need an oil change is 0.25, the probability that it needs a new

filter is 0.40 and the probability that both the oil and filter need

changing is 0.14.

a) If the oil had to be changed, what is the probability that a new

filter is needed?

b) If a new oil filter is needed, what is the probability that the oil has

to be changed?

3. A recent survey asked 100 people if they thought women in

the armed forces should be permitted to participate in

combat. The results of the survey are shown.

Gender Yes No Total

Male 32 18 50Female 8 42 50Total 40 60 100

Find these probabilities.

a) The respondent answered yes, given that the respondent

was a female.

b) The respondent was a male, given that the respondent

answered no.

4. A committee is composed of six democrats and five republicans.

Three of the democrats are men and three of the republicans are

men. If a man is chosen for chairman, what is the probability that

he is a republican?

5. A coin is tossed 3 times. Find the probability that all three are

heads.

A. If it is known that the first is head

B. If it is known that the first 2 are heads

C. If it is known that 2 of them are heads

6. A survey was made of 100 customers in a dept. store. Sixty of

the 100 indicated they visited the store because of the

newspaper ads. The remainder has not seen the ads. A total of

40 customers made purchases, of these customers 30 had seen

the ads. What is the probability that a person did not see the ad

made a purchase? What is the probability that a person who saw

the ads made a purchase?

7. A coin is tossed three times and 2 heads and 1 tail fall: what

is the probability that the first toss was a head?

8. The probability that the doctor correctly diagnoses a

particular illness is 0.70. given that the doctor makes an

incorrect diagnosis, the probability that the patient enters a

lawsuit is 0.90. what is the probability that the doctor makes

an incorrect diagnosis and the patient sues?

9. A real state agent has 8 master keys to open several new

homes. Only one master key will open any given house. If

40% of these homes are usually left unlocked, what is the

probability that the real estate agent can get into a specific

home if the agent selects 3 master keys at random before

leaving an office?

10.The probability that Tom will be alive in 20 years is 0.70 and the

probability that Nancy will be alive in 20 years is 0.90, what is

the probability that neither will be alive in 20 years?

11.The probability that a vehicle entering the Luray Caverns has

Canadian license plates is 0.12; the probability that it is a

camper is 0.28; and the probability that it is a camper with

Canadian plate is .09. What is the probability that;

a) A camper entering the Luray Caverns has canadian license

plates?

b) A vehicle with canadian license plates entering the Luray Caverns

is a camper?

c) A vehicle entering the Luray Caverns does not have canadian

license plates or is not a camper?

Statistics: Bayes' Theorem (aka, Bayes' Rule)

Bayes' theorem (also known as Bayes' rule) is a useful tool for

calculating conditional probabilities. Bayes' theorem can be stated

as follows:

Bayes' theorem. Let A1, A2, ... , An be a set of mutually exclusive

events that together form the sample space S. Let B be any event

from the same sample space, such that P(B) > 0. Then,

P( Ak | B ) = P( Ak ∩ B )

P( A1 ∩ B ) + P( A2 ∩ B ) + . . . + P( An ∩ B )

P( Ak | B ) = P( Ak ) P( B | Ak )

P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 ) + . . . + P( An ) P( B | An )

1. In a certain plant, three machines, A, B, C makes 30%, 45%, 25%,

respectively of the products. It is known that from past experiences

that 2%, 3% and 2% of the product made by each machine,

respectively are defective. Now, suppose that a finished product is

randomly selected,

a) what is the probability that it is defective.

b) If a product was chosen and found out to be defective. What is the

probability that it was made by machine C?

Answer:

Make an assumption:

Let D = defective products

A = product for machine A

B = product for machine B

C = product for machine C

2. Ana holds a box of 3 red and 2 white balls and Benjie has a box of 5

red and 2 white balls. We play a game by tossing a coin. If head came

up, we pick a ball at Ana’s box; If tails, we pick a ball from Benjie’s

box.

a) Find the probability that a white ball is chosen.

b) Suppose that it turned out to be white. What is the probability that it

came from Ana’s box?

3. Marie is getting married tomorrow, at an

outdoor ceremony in the desert. In recent years,

it has rained only 5 days each year.

Unfortunately, the weatherman has predicted

rain for tomorrow. When it actually rains, the

weatherman correctly forecasts rain 90% of the

time. When it doesn't rain, he incorrectly

forecasts rain 10% of the time. What is the

probability that it will rain on the day of Marie's

wedding?

4. A machine produces defective parts with three different

probabilities depending on the state of repair. If the

machine is in good working order, it produces defective

parts with probability 0.02. If it is wearing down, it

produces defective parts with probability 0.1. If it needs

maintenance, it produces defective parts with

probability 0.3. The probability that the machine is in

good working order is 0.8, the probability that it is

wearing down is 0.1, and the probability that it needs

maintenance is 0.1. Compute the probability that a

randomly selected part will be defective.

5. Jar 1 has 2 white and 3 green balls, Jar 2, 4 white and 1 green and

jar 3, 3 white and 4 green. A jar is selected at random and a ball drawn

at random and it is found to be white. Find the probability that Jar 1 was

selected.

6. A production engineers knows that 5% of all the PCB’s they

manufactured are defective, 92% of all the PCB’s that are defective

are also rated defective by their QC department and 2% of all the

PCB’s that are not defective are rated defective by the QC dept. What

is the probability that a PCB that are rated defective by a QC is

actually defective?

7. There are 3 urns A, B and C each containing a total of 10

marbles of which 2, 4 and 8 respectively are red. A pack of

cards is cut and a marble is taken from one of the urns

depending on the suit shown - a black suit indicating urn A, a

diamond urn B, and a heart urn C. What is the probability a red

marble is drawn?

8. Of all the smokers in a particular district, 40% prefer brand A

and 60% prefer brand B. Of those smokers who prefer brand A,

30% are females, and of those who prefer brand B, 40% are

female. What is the probability that a randomly selected smoker

prefers brand A, given that the person selected is a female?

Seatwork:

1.A class in physics is comprised of 10 juniors, 30 seniors and 10 graduate

students. The final grades show that 3 of the juniors, 10 of the seniors and 5

of the graduate students received an A for the course. If a student chosen at

random from this class and is found to have earned an A, what is the

probability that he or she is a senior? (2.78)

2.The probability that a head of household is home when a telemarketer

representative calls is 0.40. Given that the head of the household is home,

the probability that goods will be bought from the company is 0.30. Find the

probability that the head of the house is home and goods being bought from

the company. (2.88)

3. In an experiment to study the relationship of hypertension and smoking

habits, the following data are collected for 180 individuals; (2.80)

Non Smokers Moderate smokers Heavy smokers

Hypertension 21 36 30

Non-Hypertension 48 26 19

If one of these individual is selected at random, find the probability that the

person is:

a) Experiencing hypertension, given that the person is a heavy smoker

b) Non smoker, given that the person experiencing no hypertension

4. The probability that a married man watches a certain TV show

is 0.40 and the probability that a married woman watches the

show is 0.50. The probability that a man watches the show given

that his wife does is 0.70. Find the probability that: (2.85)

a)A married couple watches the show

b)A wife watches the show given that her husband does

c)At least 1 person of a married couple will watch the show

Party In Favor Not FavorRepublican 98 54Democrat 79 29

•A political telephone survey of 260 people asked whether they were in favor or not in favor of the proposed law. Each person was identified as republican and democrat. A person from the survey is selected at random.

1. Determine the probability that the selected person is in

favor of the new law

2. Determine the probability that the selected person is a

Republican

3. Determine the probability that the selected person is not

in favor of the new law


4. Determine the probability that the selected person is a

democrat


favor of the new law given that the person is a Republican

6. Determine the probability that the selected person is not in

favor of the new law given that the person is a Republican


favor of the new law given that the person is a Democrat


favor of the new law and the person is a Republican


9. Determine the probability that the selected person is in favor

of the new law and the person is a Democrat


of the new law or the person is a Republican


of the new law or the person is a Democrat

12. Using Baye’s rule, calculate the probability that the selected

person was a republican, given that the person was in favor

of the new law

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