Probability Terms and Notation - Mr....

Probability Terms and NotationTerm Definition Example Trial

Is any operation whose outcome

__________be predicted with certainty

A coin toss

Experiment Consists of ________________trials

Experimental

Probability

Event of A is represented as

P A( ) = n A( )where n(A) is the

n T( )

number of time event A occurred

and n(T) is the total number of

trials, T, in the experiment.

When a coin was tossed it

landed heads up four times

out of ten. 2

10 5P heads( ) = 4

=

Outcome The result of carrying out an

experiment

Sample space (S) The set of all possible outcomes e.g. coin: H, T

die: 1, 2, 3, 4, 5, 6

_________of the sample space. It

consists of one or more of the

possible outcomes.

e.g. coin: H

die: even numbers

Theoretical

Probability

Event of A is represented as

P A( ) = n A( ) where n(A) is the

n S( )

number of favourable outcomes for

event A and n(S) is the total number

of outcomes in the sample space, S,

where all outcomes are equally

likely.

The probability of a coin

landing heads up is

P H( ) = 12

Impossible

Probability

The event X does not include any of

the outcomes in the sample space

NOTE: Unless specified otherwise, solutions for probability are to written as reduced fractions.

Math 3201 Chapter 3: Probability Handouts

Section 3.1 Lesson 1 1

• If an experiment has a set of equally likely possible outcomes then the probability of a particularevent A is given by the formula;

P(A) = number of outcomes favourable to Atotal number of possible outcomes eg. P(Heart) =

• If the event X does not include any of the outcomes in the sample space, then the event X isimpossible and we write P(X) = 0.eg. P(rolling a 7 on a standard die) = 0

• If the event Y includes all of the outcomes in the sample space, then the event Y is certain andP(Y) = 1.eg. P(rolling a natural number less than and equal to 6 on a standard die) = 1

14

Answer the following components associated with the following scenarios.

Example: A fair die is rolled. What is the probability of rolling a “three”?

a. The sample space:

__________________________________________

b. The outcomes of the sample space favorable to the event indicated :

__________________________________________

c. Whether the outcomes are equally likely :

__________________________________________

d. The probability of the event:

__________________________________________

Example: A circular spinner is divided into four equal sectors, shaded black, red, blue, and

yellow. When it is spun what is the probability of it landing on “red”?


__________________________________________


__________________________________________


__________________________________________


__________________________________________



Example: Two coins are thrown and the number of tails is counted. What is the

probability of obtaining “two tails”?


__________________________________________


__________________________________________


__________________________________________


__________________________________________

Example: An experiment consists of rolling a six sided die, flipping a coin, and spinning a spinner divided into five regions. The number of elements in the sample space of this experiment is

A. 3 B. 7 C. 13 D. 60

Example: The probability that it will be raining in the city’s west side today is 0.67. The probability that it will not be raining is

A.0.33B. 0.52 C. 0.67 D. None of the above

Example: A 5-sided red die has two sides labelled with a 1, one side labelled with a 2 and two sides labelled with a 3. A green die is 5-sided and each of the sides is labelled with 1, 2, 3, 4, and 5 with each number used only once. The sample space for rolling each of these dice once would not include a

A. 1 on the red die and a 2 on the green die B. 2 on the red die and a 3 on the green die C. 3 on the red die and a 4 on the green die D. 4 on the red die and a 5 on the green die



Note • For any event A, 0 £ P(A) £ 1.• The notation P(A' ) means the probability of “not A,” the complement ofA, i.e. P(not A) = P(A' ) = 1 - P(A)

Example: For each of the following state: i ) the sample spacei i ) the outcomes in the sample space which are favourable to the event indicatediii)whether the outcomes are equally likelyiv) the probability of the event

a. A fair die is rolled. What is the probability of rolling a “two” ?

b. A circular spinner is divided into four equal sectors, shaded red, blue, green and yellow. When itis spun what is the probability it lands on “green”?

eg. P(rolling a 5) = 1 ,6

then P(not rolling a 5) = 1 - 1 = 56 6

The formula P(A) = 1 - P(A') is on the formula sheet.


Section 3.1 4Lesson 2

Compound Events

Events formed from repeated trials or from a combination of simple events are called compound events and often a table, a chart or a tree diagram is useful in determining the sample space.

Example: Consider an experiment of rolling an equally spaced circular spinner numbered 1 to 4 and tossing two coins.

a) Draw a tree diagram to show all the outcomes for the experiment. List the sample space.

c) Are all the outcomes equally likely?

d) State the probability of obtaining :i ) a four and two heads i i ) a prime number and exactly one tail

Example: A die one and a die two are rolled. The outcome “3 on the die one and 5 on the die two” can be represented by the ordered pair (3, 5).

1111 21

1

2

3

4

5

6

One Die

Two Die

3, 5

a) Show all the possible outcomes in the array.

b) How many points are in the sample space?

c) List the event “the same number appearson both dice” as a subset of the samplespace.

d) State the probabilities of the following events:i) the same number appears on both dice ii) a different number appears on each die

3 4 5 6

b) How many elements in the sample space? Use the fundamental counting principle to determine theanswer?



Example: A card is drawn from a standard deck of 52 cards. Use formulas to determine the probability that:a) A ace of diamonds or a spade is drawn. b) A ace or a spade is drawn.

Example: 200 people bought cars, which include sun roof and air conditioning. The results were as follows:• 60 people got a sun roof• 126 people got air conditioning

• 36 people got both

What is the probability that a person who purchases a car gota) at least one of the two options? b) neither of the options?



Jonathan Mauger

Stamp

Probability and Odds

Odds in Favour

The ______________ of the probability that an event ______ occur to the probability of the

event that the event ____________ occur, or the ratio of the number of __________________

outcomes to the number of __________________ outcomes.

( )( ')

P AP A

, or ( ) : ( ')P A P A

Odds Against

The ratio of the __________________ that an event ____________ occur to the probability that

the event ______ occur, or the ratio of the number of __________________ outcomes to the

number of__________________ outcomes.

( ')( )

P AP A

, or ( ') : ( )P A P A

Example: The odds in favour of an event happening are given. Determine the probability that each event will happen.

a. 3 : 5 ________________________________

b. 7 : 5 ________________________________

c. 100 : 1 ________________________________Example: The odds against of an event happening are given. Determine the probability that each event will happen.

a. 4 : 7 ________________________________

b. 10 : 9 ________________________________

c. 2 : 1 ________________________________



Example: The probability that an event will happen is given. Determine the odds against each event is happening.

a. P(A) = 45%________________________________

b. P(C)= 12%________________________________

c. P(D)= 75%________________________________

Example: Two brown-eyed people are told that there is a 35% probability that their baby will have blue eyes. What are the odds in favour of such parents having a baby with blue eyes? What are the odds against?

Step 1

Step 2

Step 3

Step 4

Example: Max has gone to the store to buy a pair of jeans. From experience, he knows the odds against the store having his style of jeans in her size are 10 : 32. Determine the probability that the store will have jeans in his size.

Step 1

Step 2



Example: Ratings for the movie Butterfly Dreaming indicate that 45% of the viewers are male, 10% are under 18, 20% are 19-29 years old, 20% are 30 to 45 years old, and 50% are older than 45. Suppose that someone watching Butterfly Dreaming.

a. What are the odds in favor of this person being female?

b. What are the odds against this person being 45 or younger?

Example: Laser printers are on sale at Staples. The last 9 times laser printers were on sale, they were in stock only 4 times.

a. Determine the odds in favour of laser printers being in stock this time.

b. Determine the odds against laser printers being in stock at this time.



Probabilities Using Counting Methods

For this lesson you need to remember…

• Use _____________________ when order is important

• Use _____________________ when order is not important

Example: A committee of five people is selected from ten females and eight males.What is the probability that there are exactly three females on the committee?

What is the probability that there are exactly three males on the committee?

What is the probability that there are exactly four females on the committee?

Example: Three people line up to buy a ticket at the movie theatre.

What is the probability that they line up in descending order of age?

What is the probability that they line up in ascending order of age?



Example: Steven, Brittany, Julie, and Max are volunteering along with five other students on their school’s math team. All the students have equal ability. Determine the probability that Steven, Brittany, Julie, and Max will be chosen to fill the five spots on the team.

Example: In the card game Crazy Eights, players are dealt 8 cards from a standard deck of 52 playing cards. Determine the probability that a hand will contain exactly 7 hearts.

Example: Dar spells out COOKBOOK with letter tiles. The tiles are face down and mixed up. He asks Devon to arrange the tile in a row and turn them face up. If the row of tiles spells COOKBOOK, Devon will win a series of gift cards for local restaurants. Determine the probability that Devon will win.

Example: Access to a particular online site is password protected. Every member must create a password that consists of 3 capital letters followed by 2 digits. For each condition below, determine the probability that a password chosen at random will contain the letters A, B, and C.

a. Repetitions are not allowed in a password.

b. Repetitions are allowed in a password.



Probability Involving Permutations & Combinations

Using Permutations or Combinations to Find the Probability of an Event Example: Two cards are picked without replacement from a deck of 52 playing cards. Determine the probability that both are Jacks using

a) the multiplication law b) combinations

Example:The word MATHEMATICS has been spelled using letter magnets. Two letter magnets are randomly chosen one at a time and placed in the order in which they were chosen. Determine the probability that the letter magnets are:

a) MA b) both vowels

Example: The School Council decides to form a sub-committee of six council members to look at how funds raised should be spent on safe grad activities. There are a total of 16 school council members, 7 males and 9 females. What is the probability that the sub-committee will consist of exactly 3 males?

Example: A bag of candy contains 6 red, 4 green, and 5 yellow candies. If a kid grabs three candies from the bag, determine the probability that:

a) exactly 2 are green b) at least one is green

c) d)the first is red, the second isgreen and the third is yellow

one is red, one is green and one is yellow


Section 3.3 Lesson . 12

Example: School Scholarship Council consists of five men and ten women. Three representatives are chosen at random to form a ESL student of the year sub-committee.

a) What is the probability that Princple Barry and two women are chosen?

b) What is the probability that two women are chosen if Princple Barry must be on thecommittee?

Example: In a game of crib you have 4 cards in your hand and one cut on the deck from a pack of 52 shuffled cards.When you look at your 5 cards, what is the probability, expressed in combination notation, that you have:

a) four fives? b) four fives and a jack?

c) 10, J, Q, K and five? d) at least one five?

Example: In a class of 20 students, calculate the probability (to the nearest hundredth) that:

a) they all have different birthdays (assume no one is born on February 29)

b) at least 2 of them have the same birthday.



Mutually Exclusive Events and the Event “A or B”

The Events “A or B”, “A and B”

The event A or B is said to occur if the event A occurs or if the event B occurs or if both events occur.The event A and B occurs if both event A and event B occur simultaneously.

Example: Consider the experiment of rolling a die. Let the event A be “an even number is thrown” and the event B be “an odd number is thrown”.

A B

a) Complete the Venn Diagram which represents the sample space.

b) List the outcomes for:

i ) the event A i i ) the event B

iii) the event A or B iv) the event A and B

c) Complete the following:

n(A) = ____ n(B) = ____ n(A or B) = ____ n(A and B) = ____

d) Determine the following probabilities:

P(A) = ____ P(B) = ____ P(A or B) = ____ P(A and B) = ____

• Notice that the events A, B have no common outcomes.

The events A, B are called ________________________

• Notice that in the Venn Diagram the circle for A and the circle for B have no area of overlap.

Rule: P(A or B) = P(A) + P(B)

•

•



Example: Consider the experiment of rolling a die. Let the event A be “an odd number is thrown” and the event B be “a multiple of three” is thrown.

A B

a)

b) List the outcomes for:i ) the event A i i ) the event B

iii) the event A or B iv) the event A and B

c) Complete the following:

n(A) = ____ n(B) = ____ n(A or B) = ____ n(A and B) = ____

d) Determine the following probabilities:

P(A) = ____ P(B) = ____ P(A or B) = ____ P(A and B) = ____

• Notice that the events A, B have common outcomes.

The events A, B are not _________________________.

• Notice that in the Venn Diagram the circle for A and the circle for B have an area of overlaprepresenting the event A and B.

Rule: P(A or B) = P(A) + P(B) - P(A and B)

Mutually Exclusive Events

Events are said to be mutually exclusive if they have no common outcomes.

Probability of the Event A or BUse the following formulas for the probability of the event A or B.

If the events A, B, are mutually exclusive then

P(A or B) ==== P(A) ++++ P(B)

If the events A, B, are NOT mutually exclusive, then

P(A or B) ==== P(A) ++++ P(B) ---- P(A and B)

Complete the Venn Diagram which represents the sample space.



Example: Consider drawing a card from a standard deck. The following events are defined:

• Event A - a face card is selected• Event B - a ace selected• Event C - an heart is selected• Event D - a black card is selected

State all the pairs of events which are mutually exclusive.

A B

C

D

Use the Venn Diagram to state all the pairs of events which are mutually exclusive.

Use the following information to determine whether the events A, B are mutually exclusive.14

13P(A) = P(B) = P(A or B) =

712

S H

45% 18% 27%

10%

Example: A baseball team was surveyed to find out whether they did stretch or hydrate before the game. The Venn diagram shows the percentage of players in each category.

If a player is selected at random from the team determine the probability that the student:

a) Stretch b) Stretch and Hydrate c) Stretch or Hydrate



Conditional Probability and the Event “A and B”

Dependent Events

Two events are dependent if the knowledge that one event has occurred changes the probability of the other event occurring.

Example: One card is drawn from a deck of cards and is replaced. A second card is then drawn. Consider the following eventsA = the first card is a spadeB = the second card is a spadea) Determine P(A) b) Determine P(B)

The probability of event B does NOT depend on whether or not event A occurred. Events A and B are called independent events.

Independent Events

Two events are independent if the knowledge that one event has occurred has no effect on the probability of the other event occurring.

Example: Classify the following events as dependent or independent.a) The experiment is rolling a die and tossing a coin.

The first event is rolling 3 on the die and the second event is tossing heads on the coin.

b) The experiment is choosing two cards without replacement from a standard deck.The first event is that the first card is a queen and the second event is that thesecond card is a queen.

c) The experiment is choosing two cards with replacement from a standard deck.The first event is that the first card is a jack and the second event is that thesecond card is a jack.

•

•

•



Conditional Probability

One card is drawn from a deck of cards and is not replaced. A second card is then drawn. Consider the following events.

A = the first card is a spade B = the second card is a spade

P(A) = 1352 =

14 but P(B) depends on whether the event A occurred or did not occur.

• We denote by P(BΩA) the conditional probability of the event B, given that the event Ahas occurred.In this example P(BΩA) = _____

• We denote by P(BΩA) the conditional probability of the event B, given that the event Ahas NOT occurred.In this example P(BΩA) = _____

Die Two

One Die

111 221

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

Example: A die one and a die two are tossed. The outcomes are shown in the array. Consider the following two events:

Event A the sum of the two dice is 6

Event B the number on each die is the same

a) List the outcomes for the following events as subsets of the sample space.

i ) A =

i i ) B = iii) A and B =

b) State the following probabilities:i ) P(A) = i i ) P(B) = iii) P(BΩA) =

iv) P(AΩB) = v) P(A and B) = vi) P(B and A) =

c) Demonstrate the following:

i ) P(A and B) = P(A) ¥ P(BΩA) i i ) P(A and B) = P(B) ¥ P(AΩB)



Multiplication Law for Dependent Events

Given that two events A, B, are dependent, then

P(A and B) ==== P(A) ¥¥¥¥ P(BΩA)

Example: Two cards are drawn without replacement from a standard deck of 52 cards. Determine the probability of the following events:a) both cards are black b) neither card is a heart

c) d)the first card is a queen and the second card is a six.

one of the cards is a jack and the other is a ace.

Multiplication Law for Independent Events

If the events A, B, are independent, then the knowledge that event A has occurred has no effect on the probability of the event B occurring.

Thus P(BΩA) = P(B).Arriving at the following law for independent events:Given that two events A, B, are independent, then

P(A and B) ==== P(A) ¥¥¥¥ P(B)

Example: Two cards are drawn with replacement from a standard deck of 52 cards. Determine the probability of the following events:a) both cards are black b) the first card is a queen and the second card is a six.

c) one of the cards is a jack and the other is a ace.



Mutually exclusive events and independent events are mean the same thing.

• The concept of mutually exclusive events, involves whether or not two events can occursimultaneously.

• The concept of independent events involves whether or not the occurrence of one eventhas an effect on the probability of the other event occurring.

Let A and B be events with P(A) = 12 , P(B) =

13 , and P(A and B) =

14 . Find:

a) P(AΩB) b) P(BΩA) c) P(A or B)

Example: The probability that Ajay will hit a homerun during tonights game is 0.7 and the probability that he will strike out during tonights game is 0.9. If these events are independent, what is the probability (to the nearest hundredth) that he will a) hit homerun and strike out b) hit homerun or strike out

c) hit homerun but not strike out d) neither hit a homerun nor strike out

Example:



Probability Problems Involving Independent EventsIf the events A, B, are mutually exclusive, then P(A or B) = P(A) + P(B)

If the events A, B, are NOT mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B)

If the events A, B, are independent, then

P(A and B) = P(A) ¥ P(B)

If the events A, B, are dependent, then

P(A and B) = P(A) ¥ P(BΩA)

Example: Two hockey players, Sydney and Phil, each independently take a penalty shot.

Sydney has a 810 chance of scoring, and Phil has a chance of scoring. What is the

probability that;a) both score? b) both miss?

c) only one of them scores? d) at least one of them scores?

Example: The score at the end of regular time was Toronto 2 Montreal 2. After three shootout shots for each team in the first round, the game was still tied. The game will now be decided by a sudden death shootout where each team takes alternate shots on goal. Each team shoots once in each round. If both teams score, or both teams miss, they go on to another round. From past records the probability that Toronto will score on a shootout shot is 0.7 and the probability that Montreal will score on a shootout shot is 0.6. What is the probability that;a) Toronto wins in the second round b) Montreal wins in the second round?

c) Toronto wins in the third round?

35

•

•

•

•



Probability: Tree Diagram for Independent Events

Example: Beatrice is planning a family BBQ during the up coming holiday weekend (3 days). She is considering cancelling the BBQ if rains. The probability of a rain is 0.2 and the weather on each day is independent of the weather on the other days.

0.2

0.8

0.2

0.2

R1

R2

R1

R2

R2

R3

R2

P(R1 and R2 and R3) =

Complete the tree diagram to determine the probability that;a) it will rain all 3 days

b) there will be only 1 day of rain

c) that at least 2 days will be dry



Probability: Infinite Geometric Sequences

Example: Consider the following situation.

Two siblings (Jerry and Butch) are trying to decide who gets to drive the family car on Friday night. To decide who drives the car, they roll a die alternately until one of them rolls a one. Jerry rolls first.

Find the probability that Jerry drives the family car.

Let Ji be the event that Jerry rolls a one on his ith attempt.

Let Bi be the event that Butch throws a one on his ith attempt.

Jerry will drive the family car if any one of the following events occurs:

Jerry throws a one on his first attempt

neither Jerry nor Butch throws a six on their first attempt and Jerry throws a six on his second attempt

neither Jerry nor Butch throws a six on their first two attempts and Jerry throws a six on his third attempt

P(J1)

Event Probability Notation

P(J1 and B

1 and J2)

Probability

16

ÊÁÁ Ë

56

ˆ

2

¥ 16

56 ¥

56 6 ¥

1 =

or

or

or

The probabilities in the extreme right column form an infinite geometric sequence.

a) Complete the next row of the table and use the results to determine the common ratio ofthe infinite geometric sequence.

b) Find the sum of the terms of this infinite geometric sequence to determine the probabliltythat Jerry starts the game.



Jonathan Mauger

Text Box

Not a Course Objective

Probability Problems Involving Conditional ProbabilityReview

If the events A, B, are mutually exclusive, then P(A or B) = P(A) + P(B)

If the events A, B, are NOT mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B)

If the events A, B, are independent, then P(A and B) = P(A) ¥ P(B)

If the events A, B, are dependent, then

P(A and B) = P(A) ¥ P(BΩA)

The formula for dependent events can be written as P(AΩB) = P(A and B)

P(A)

Example: One card is drawn at random from a deck of 52 cards. The following events are defined:

A: a spade is drawn B: a ace is drawn C: a black card is drawn

Express the following probabilities as fractions in simplest form:

a) P(A) = _____ b) P(A or C) = _____ c) P(B and C) = _____

d) P(A and B) = _____ e) P(AΩC) = _____ f) P(CΩA) = _____

Example: Two fair dice are rolled. Calculate the probability that 2 “threes” are rolled given that at least 1 “three” is rolled.

• •

••

•



BusB

CarC

OtherO Total

Male, M

Female, F

Total

350350 75

300300 100

Example: The table shows how the students in a large high school generally travel to school.

a) How many students attended the high school?

c) If a student is selected at random, determine the probability thati ) the student is male i i ) the student travels by bus i i i) the student is male and travels by bus

d) Determine the probability that:i ) a female student travels by bus. i i ) a student who drives is male

e) Are the events “the student is female” and “the student travels by bus” independent events?Explain.

Example: Todd naviagting a local hay maze, the probability he turns right the next time is increased by 25%. If Todd turns left, the probability he turns left the next time is decreased by 25%. Assuming that there is an equal probability that the first turn will be to the left or right, calculate the probability that:

a) the first two turns are both right b) the first two turns are both left

c) the first two turns are different d) the first three turns are all left

100

75



Using a Probability Tree For Dependent Events

Step 1: Introduce symbols to represent the information.

Step 2: Write the given probabilities in terms of the symbols.

Step 3: Set up a tree diagram.

a) Answer the following

• M1 - bottle is from machine M1. M2 - bottle is from machine M2. B - bottle is broken.

• P(M1) = P(M2) = P( ) = 0.06 P( ) = 0.09

b) Complete the probability tree diagram, where the first branches leading to the machines andthe second set of branches leading to the defective/non-defective items.

A1

B

P(M1) = 0.6

P(M2) =M2

B

P(BΩM1) = Æ P(M1 and B) = P(M1) P(BΩM1)

P(B ΩM1) = B ÆP(M1 and B ) =

c) i ) If a bottle is chosen at random determine the probability that the bottle is broken.

i i) Write a formula for P(B) in terms of conditional probabilities.

Example: Two machine M1 and M2 produce all the light bulbs in a factory. Machine M1 produces 60% of the output. The percentages of broken bulbs produced by these machines respectively are 6% and 9%.Method

B Æ P(M1 and B ) =



Jonathan Mauger

Line

Jonathan Mauger

Line

In general if a sample space is partitioned into mutually exclusive outcomes M1, M2, M3, M4 . . . and if B is any other event then

P(B) = P(M1 and B) + P(M2 and B) + P(M3 and B) + P(M4 and B) + . . .

P(B) = P(M1 ) P(BΩM1 ) + P(M2 ) P(BΩM2 ) + P(M3 ) P(BΩM3 ) + P(M4 ) P(B| M4 ) + . . .

Example: Bag A contains 6 red and 4 green marbles. Bag B contains 8 red and 2 green marbles. One of the bags is chosen by selecting one card at random from a deck of cards. If a spade is selected, then a marble is taken at random from Bag A. If a spade is not selected, then a marble is taken from Bag B.

What is the probability that the marble is green?”

a) Complete:P(A) = P(B) = P(G ΩA) = P(G ΩB) =

b) Rewrite the formula on total probability using the symbols in a) and solve the problem.

c) Use a tree diagram to solve the problem.



Example: Two boxes contain 8 dinky cars. The first box contains 4 blue dinky cars and 4 red dinky cars. the second box contains 2 blue dinky cars and 6 red dinky cars. One of the boxes is seletced at random and a dinky car is removed. It is a red dinky car. A second dinky car is them removed from the same box.

Calculate the probability that it is a red dinky car.



Bayes’ LawIf a sample space is partitioned into mutually exclusive outcomes A1, A2, A3 . . .and if B is any other event then

P(A1ΩB) = P(A1 and B)

P(B) = P(A1 ) P(BΩA1 )

P(A1 ) P(BΩA1 ) + P(A2 ) P(BΩA2 ) + ...

Example: Box A contains 5 green balls and 3 white balls. Box B contains 6 green balls and 4 white balls. A box is selected at random and a ball is drawn from that box. if the ball is green, what is the probability it came from box A?

a) a person reacts positively to the test.

b) a person showing a positive reaction is actually pregnant.

Example: A new test for detecting pregnancy was developed. Medical trails show that 93% of the patients who were pregnant tested positively to the new test, while 4% of patients who were not pregnant tested positive. If 5% of the population are pregnant. Determine the probability that:

Example: A new test for detecting prostate cancer has been developed. Medical trials show that 92% of the patients who have the disease react positively to the new test, while 5% of patients not suffering from the disease also react positively. If 4% of the population have the disease, determine the probability that a person testing positive actially has prostate cancer.



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