P(A)=2/3, P(B)=3/5, And P(AB)=11/15, Find P(B|A) (2024)

Functions of the form f(x)=kx represents direct variation.

The table of values that appears a constant proportion between two variables speaks to a direct variation function.

A direct variation is an equation in the form y = kx, where k is the constant of variation. Since it is in this form, the equation will pass through the origin (0, 0); this is because if we use 0 as x, y = k(0) = 0.

In the event that there's at slightest one pair of values that features a different proportion, at that point the function isn't a direct proportion. the equation applies to each corresponding value between the two variables. Functions of form f(x)=kx, where k is the constant, are as they were functions that can speak to a direct variation. T

Therefore, A direct variation is an equation in the form y = kx, where k is the constant of variation. Since it is in this form, the equation will pass through the origin (0, 0); this is because if we use 0 as x, y = k(0) = 0.

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To calculate the interest and the total amount , we use Simple interest formula S.I = ( P× r × t ) / 100.

Simple interest is a method of calculating the interest that is charged on fixed deposits , savings accounts and the loans .

The money you put in the bank is called the principal, the bank pays you interest.

The interest is computed as a certain percent of the principal; called the rate of interest. The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent.

The variable for time, t, represents the number of years the money is left in the account.

The formula to calculate simple interest is :

S.I = ( P× r × t ) / 100

where P is principle , r = rate of interest and t is term of loan in year.

To use the simple interest formula we substitute in the values for variables that are given, and then solve for the unknown variable.

To find the total amount ,

Amount = Principle + Interest

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The required relation on R on A is explain below

What is finite non-empty set?

It is a set where either the quantity of components is large or just beginning or finishing is given. In this way, we mean it with the quantity of components, n(A), and on the off chance that n(A) is a characteristic number, it's a limited set.

According to question:

Let A be finite non-empty set.

Then the exist relation R on A which is both equivalence and order relation.

Suppose A = {a₁, a₂, a₃........an}

And, R = {(a₁, a₁)(a₂, a₂).......}

R not equal to ∅ and it is reflexive, symmetric, transitive, anti symmetric.

Then R is equivalence relation and an order relation.

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The number of different possible orders that are there for the group is 15504 .

In the question ,

it is given that ,

total number of friends = 5 ,

the number of dishes that restaurant offers for dinner = 20 dishes .

It is given that, group collectively orders 5 different dishes to share and waiter just needs to place all 5 dishes in the center of the table.

So , the 5 dishes from the 20 dishes can be selected in ²⁰C₅ ways .

²⁰C₅ = 20!/(15! * 5!)

= 20*19*18*17*16*15! / (15! * 5!)

= (20*19*18*17*16)/(5*4*3*2*1)

Simplifying Further , we get

= 1860480/120

= 15504 ways

Therefore , There are 15504 possible orders for the group .

The given question is incomplete , the complete question is

A group of five friends go to a restaurant for dinner. the restaurant offers 20 different main dishes. Suppose that the group collectively orders five different dishes to share. the waiter just needs to place all five dishes in the center of the table. how many different possible orders are there for the group ?

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The equation for the direct variation will be 5y-3x=0.

What is the equation?

An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.

It is given that, For the data in the table, does y vary directly with x

X: 5, 8, 9

Y: 3, 5, 7

y ∝ x

y=kx

3=k5

k=3/5

For x as 5 and y as 3, the obtained equation is,

y=kx

y=3/5x

5y=3x

5y-3x=0

Thus, the equation for the direct variation will be 5y-3x=0.

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In this scenario, the dependent variable is the time the pencil takes to become dull and the type of pencil.

What is a function?

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

It is given that, there are three different pencils that Monique possesses. She's interested in finding out what kind of pencil maintains its sharpness the longest. Each pencil she uses leaves an identical mark on a piece of paper.

There are often two different sorts of variables in analytics. We obtained that independent variables will have an impact on dependent variables. What occurs as a result of the independent variable is referred to as a dependent variable.

Thus, in this scenario, the dependent variable is the time the pencil takes to become dull and the type of pencil.

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Answer:

3 : 2

Step-by-step explanation:

the ratio of

length : breadth

= 15 : 10 ( divide both parts by 5 )

= 3 : 2

By using the power property, the resultant answer when we rewrite log3x9 is (B) 9log3x.

What is power property?

Multiply the exponents to determine the power of the power.

This is a development of the powers property.

Consider raising a number to a power and repeatedly multiplying the entire phrase by itself.

According to the power of a powerful formula, you can multiply two exponents together if one is increased to a power.

How many times to multiply two numbers together is indicated by an exponent.

For instance, 42 instructs you to multiply by 4 and 4. Alternatively, multiply four by two.

So, we have: log3x9

We have the following for logarithm properties:

loga (x ^ b) = b * loga (x)

As a result of this property, in this instance, we have:

log3 (x ^ 9) = 9log3 (x)

Therefore, by using the power property, the resultant answer when we rewrite log3x9 is (B) 9log3x.

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Correct question:
Use the power property to rewrite log3x9.

A) 3log9x

B) 9log3x

C) 3logx

D) 2 + log3x

The quadrants that give the solution of the inequality y ≤ ²/₃x - 4 is; I, III, and IV

What is the solution to the given inequality?

We are given the inequality as;

y ≤ ²/₃x - 4

Let us draft some coordinates before plotting the graph;

When x = 0;

y ≤ ²/₃(0) - 4

y ≤ -4

When x = 1;

y ≤ ²/₃(1) - 4

y ≤ -3¹/₃

When x = 2;

y ≤ ²/₃(2) - 4

y ≤ -2 ²/₃

when x = -1;

y ≤ ²/₃(-1) - 4

y ≤ -4²/₃

From the attached graph Inequality, it is clear that the shaded area which represents the quadrants that possess the solution to the inequality that;

Quadrants I, III, and IV

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Answer:

1176 oz of drink

Step-by-step explanation:

112 * 4 = 448

112 * 3 = 336

112 * 2.5 = 280

112 * 1 = 112

The sum of the first 30 terms of this arithmetic expression (A.P) will be -30.

Let the first term of the Arithmetic progression be a

And the common difference of this progression be d

An be the nth term of the progression

Sn be the sum of the first nth term of the progression

As we know,

Sn = n/2( 2a + (n-1)d)

S11 = 11/2(2a + (11-1)d)

S11 = 11/2(2a + 10d) = 11(a + 5d)

As S11 = 19

Therefore, 19 = 11(a + 5d)

19 = 11a + 55d………….(i)

Again, S19 = 11 = 19/2 ( 2a + 18d )

11 =19( a + 9d)

11 = 19a + 171d………(ii)

Now, multiplying equation (i) by 19 and (ii) by 11 and then subtracting them, we get

361 = 209a + 1045d

121 = 209a + 1881d

240 = -836d

d = -240/836 = -60/209

Putting the value of d in equation (i), we get

11a - 55x60/209 = 19

11a = 3300/209 + 19

11a = 7271/209

a = 661/209

Therefore sum of first 30 terms are S30 = 30/2(2(661/209)+29(-60/209))

S30 = 15(1322/209 -1740/209) = 15 (-418/209) = 15(-2) = -30

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Using Compund interest formula, we can say that total Rs. 67 lakh and sixty thousand is owned by Sunil to the bank after 2 years .

Compound Interest Formulas :

You can also use the principal and interest rate to find compound interest. H. Total interest on principal and accrued interest not repaid within the repayment period. The compound interest formula is:

A = P( 1 + r/100)ᵗ

where A --> amount to pay by person after t years

P --> amount borrowed by person

r ---> rate of interest annually

t-----> time period

We have given that,

The amount borrowed by Sunil for education loan(P) = Rs. 40 lakh

Rate of interest (r) = 30% per annum

Interest is compunded annually.

time period (t) = 2 years

we have to calculate the amount to pay by Sunil after two years .

putting all known values in above formula,

A = 40,00000 ( 1+ 30/100)²

=> A = 40,00000(13/10)²

=> A = 40,00000×13×13/100

=> A = 40,000×169

=> A = 67,60,000

Hence, the amount that Sunil own to bank after two year is 67 lakh sixty thousand.

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The length of Stan's movie is 2 hours 25 minutes

The length of Talia's movie is 2 hours 55 minutes.

What are the length of the movies?

In order to determine the length of the movies, subtract the end time of the movie from the time the movie started.

Subtraction is the mathematical operation that is used to calculate the difference between two or more numbers. The sign that is used to represent subtraction is -.

Length of the movie = time the movie ended - time the movie started

Length of Stan's movie = 9:40 - 7:15 = 2 : 25 hours

Length of the Talia's movie = 10 : 10 - 7 : 15 = 2 : 55 hours

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Four electrons are required for the reduction of one molecule of oxygen to water in the mitochondria

Reduction means that a reaction in which electrons are added to a compound or molecule; the compound or molecule that gains electrons is said to be reduced

Water has 2 hydrogen and each hydrogen needs only one electron. Hence, four electrons along with four hydrogen atoms are required to reduce one molecule of oxygen (O₂) to two molecules of water (H₂0) in the mitochondria

Therefore, four electrons along with four hydrogen atoms are required to reduce one molecule of oxygen (O₂) to two molecules of water (H₂0) in the mitochondria

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When one or more of the activities that make up each business's value chain offer chances for cross-business cooperation to develop important new resource strengths and competitive capabilities, there is a strategic fit between the firms.

Given,

Strategic fit between two or more businesses exists when one or more activities comprising their respective value chains present opportunities:

We have to find the correct definition from the options given;

Strategic Fit;-

The degree to which a business is aligning its resources and talents with the opportunities in the external environment is expressed by its strategic fit. The organization must therefore have the actual resources and competencies to implement and support the plan because the matching is done through strategy.

Here,

The correct answer is Option D;- For cross-business collaboration to build valuable new resource strengths and competitive capabilities

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Question is incomplete. Completed question is given below;

Strategic fit between two or more businesses exists when one or more activities comprising their respective value chains present opportunities:

A. to prevent the transfer of expertise or technology or capabilities from one business to another.

B. to independently preserve common brand names from cross-business usage.

C. to increase costs by combining the performance of the related value chain activities of different businesses.

D. for cross-business collaboration to build valuable new resource strengths and competitive capabilities.

E. to maintain business value chain activities separate and apart from one business to another to protect company independence.

Proved that f is pose a continuous function on R. If each of the functions fn is bounded, is necessarily true that f is bounded.

Given:

Let {fn} be a sequence of continuous functions defined on R, and that fn + f uniformly on every finite interval [a, b]. Prove that f is pose a continuous function on R.

Denote by fn(x) any continuous function such that fn(x)=f(x) if x∈[−n,n], and fn(x)=0 if x∈(−∞,−n−1)∪(n+1,+∞). It is easy to see that such a function exists, and is bounded.

Now, using the fact that every interval [a,b] is contained in a interval [−N,N], you can check easily that (fn) converges to f uniformly on every [a,b] (actually, fn|[a,b] is constant equal to f|[a,b] for n sufficiently large).

No, because this would not give a continuous function. Define fn equal to f on [−n,n], and equal to zero on (−∞,−n−1)∪(n+1,+∞), and complete it as you want and the intervals [−n−1,n) and (n,n+1] to get a continuous function.

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Based on the Number of students who use mobile in a traditional class,

The joint relative frequency of students who use mobile in traditional classes is 62.162%

What is Joint Relative Frequency?

Joint relative frequency is defined as the ratio of the frequency that is not in total rows or the total columns to the frequency of total rows or columns.

According to the given information:

Courses/devices Traditional class Online class Total rows

Computer 28 62 90

Mobile device 46 64 110

Total Columns 74 126 200

Total Number of students who opted for traditional classes = 74 Number of students who use a mobile phone in traditional class = 46

Joint relative frequency of students who use mobile in a traditional class

= Number of students who use mobile in traditional class/Total

students who opted for traditional classes

= 46/74

Percentage of Joint relative frequency = 46/74 ×100

= 62.162%

Based on the Number of students who use mobile in a traditional class,

The joint relative frequency of students who use mobile in traditional classes is 62.162%

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Answer:

The cost per unit is the same for the 12 ounce bag and the 26 ounce bag. Since the cost is the same, it would come down to would you rather have 12 ounces or 26 ounces? The 8 ounce bag cost the most.

Step-by-step explanation:

Find the unit costs

[tex]\frac{3.40}{8}[/tex] = .425

[tex]\frac{4.80}{12}[/tex] = .4

[tex]\frac{10.40}{26}[/tex] = .4

If the one factor is the gcf and the other is the sum of two numbers that do not have a common factor, then the expression will be 8(7 + 12)

The expression is given by

56 + 96

First we have to find the GCF of the numbers

Prime factorization

56 = 2×2×2×7

96 = 2×2×2×2×2×3

The GCF(56, 96) = 8

Write the expression using the GCF

56 + 96 = 8(7) + 8(12)

Here we have to apply the distributive property

AB + AC = A(B + C)

Then,

8(7) + 8(12) = 8 (7 + 12)

Hence, if the one factor is the gcf and the other is the sum of two numbers that do not have a common factor, then the expression will be 8(7 + 12)

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Solution of the given expression 6a + 18ab after rewriting using a common factor is equal to :

A. 6a(1 + 3b).

As given in the question,

Given expression is equal to :

6a + 18ab

Find common factors of both the terms we get.

6 = 1, 2, 3, 6

18 = 1, 2, 3, 6, 9, 18

Highest common factor of 6 and 18 is equal to 6.

Now take 6a as a common factor from both the terms we get,

6a + 18ab

= ( 6a × 1 ) + ( 6a × 3b )

= 6a × ( 1 + 3b )

= 6a ( 1 + 3b )

Therefore, solution of the given expression 6a + 18ab after rewriting using a common factor is equal to :

A. 6a(1 + 3b).

The complete question is :

Rewrite 6a + 18ab using a common factor.

A.6a(1 + 3b)

B. 6a(0 + 3b)

C.6ab(0 + 3)

D. 6ab(1 + 18ab)

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Answer: 6a(1 + 3b)

Step-by-step explanation: I took the test

There is a total of 190 ways to share apples between them.

What is a ball-and-urn method?

The ball-and-urn method often referred to as stars-and-bars, sticks-and-stones, or dots-and-dividers is a typical combinatorial method.

It is used to resolve issues of the following type: in how many different ways can one divide 'k' undifferentiable items into 'n' distinct bins?

Use the ball-and-urn method, Let Alice get 'x' apples, Becky get 'y' apples, and Chris get 'z' apples.

⇒x + y + z = 24

We can manipulate this into an equation that can be solved using the ball-and-urn method

All of them get at least 2 apples, so we can subtract 2 from x, 2 from y, and 2 from z.

⇒(x - 2) + (y - 2) + (z - 2) = 18

Let x' = x - 2, y' = y - 2, z' = z - 2.

⇒ x' + y' + z' = 18

We can allow either of them to equal 0, hence this can be solved by the ball-and-urn method.

By the ball-and-urn method, we get

[tex]$\binom{18 + 3 - 1}{3 - 1} = \binom{20}{2} = \ 190}$.[/tex]

Hence, there are 190 ways to split the apples among them.

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Answer:

PQ = 49x = 10

Step-by-step explanation:

You have two figures showing expressions for a midline length and the parallel base length. In one case, you want the length of the midline (PQ). In the other case, you want the value of the variable (x).

A midline is the line segment that joins the midpoints of two sides of a triangle. It is parallel to the base, and half the length of the base.

PQ

2PQ = ON

2(5x -26) = 9x -37 . . . . . use the given expressions

10x -52 = 9x -37 . . . . . simplify

x = 15 . . . . . . . . . . . . . add 52 -9x

PQ = 5(15) -26 = 49

The length of PQ is 49 units.

X

2MN = KL

2(2x+7) = 7x -16 . . . . . . use given expressions

4x +14 = 7x -16 . . . . . . . . simplify

30 = 3x . . . . . . . . . . . . . . add 16 -4x

10 = x . . . . . . . . divide by 3

The value of x is 10.

The magnitude of 2(u-v +w) is 17.1.

The given vectors are u = 2i + 3j, v=-i, and w = 5i – 6j.

We need to find the value of 2(u-v+w).

How to find the magnitude of vector?

The formula to determine the magnitude of a vector (in two-dimensional space) v = (x, y) is: |v| =√(x²+ y²). This formula is derived from the Pythagorean theorem.

Now, 2(2i + 3j-(-i)+5i-6j)

=2(2i+3j+i+5i-6j)

=2(8i-3j)=16i-6j

The magnitude of 16i-6j=√16²+6²=√256+36=√292

Therefore, the magnitude of 2(u-v +w) is √292 which is 17.1.

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The possible length of the region after solving the quadratic equation is either 170ft or 50ft.

What is a quadratic equation?

A quadratic equation in algebra is any equation that can be written in the simpler form where x stands for an unknown value and a, b, and c are known quantities. The assumption is that a > 0; an equation with a = 0 is thought to be degenerate because they become linear or even simpler.

Solution Explained:
Lets assume, length = x and width = y

So we know, the area and perimeter is supposed to be 8500 and 440 respectively

we form the equations,

1) 8500 = xy , and [Area = length*width]

2) 440 = 2(x + y) [ Perimeter = 2(length +width)]

220 = x + y

So, x = 8500/y (from equation 1) and put the value of x in 2

220 = x + (8500/x)

After solving this we get the quadratic equation,

x^2 - 220x + 8500 = 0

Solving the equation, we get

x = 170 or, x = 50 which are the two possible lengths of the region.

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To calculate the covariance and correlation coefficient of U and V, we need to first find the expected value and variance of U and V.

What is covariance?

The covariance of two variables is a gauge of their relationship. In other words, it is a measure of how much the values of two variables tend to fluctuate together. Positive covariance shows that the values of the two variables tend to rise or decrease together, whereas negative covariance indicates that the values of one variable tend to increase when the values of the other variable drop. In probability theory and statistics, covariance is frequently employed to determine the association between two variables. It is determined by multiplying the variances of the two variables from their respective means and dividing the result by the total number of observations. This metric can be helpful for seeing trends and patterns in data and aiding analysts in forecasting.

How to solve?

The expected value of U is $E[U] = E[2 - X_1 - 2X_2] = 2 - m - 2m = -2m. Similarly, the expected value of V is $E[V] = E[3 + 3X_2 - X_3] = 3 + 3m - m = 4m.

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To calculate the value of covariance and correlation coefficient of U and V, we have to find their expected value along with variance.

What does covariance mean?

A measure of the link between two random variables and how much they fluctuate together is called covariance. Alternately, we may say that it establishes the relationship between the two variables' changes, i.e., that a change in one variable is equivalent to a change in the other.

What does the covariance tell us?

When one variable changes, covariance shows the link between the other two variables. Both variables are said to have positive covariance if a rise in one causes an increase in the other. Reduced values of one variable also result in reduced values of the other.

for expected value of U ;

= $E[U]

= E[2 - X_1 - 2X_2]

= 2 - m - 2m = -2m.

Similarly, the expected value of V;

= $E[V]

= E[3 + 3X_2 - X_3]

= 3 + 3m - m

= 4m.

now we can easily calculate the covariance and correlation coefficient of U,V.

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The sample proportion of the random sample of 50 students is 64% or 0.64.

Given:

the college administrator was only able to survey a random sample of 50 students.

she finds that 32 of these students report having a full- or part-time job.

we are asked to determine the sample proportion = ?

Based on the given conditions, we formulate:

32÷50

simplify:

16/25

= 64% or 0.64

hence we get the required sample proportion.

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The average book owned by other people is 112 which is greater than Michael's.

What is average?

In standard language, a mean could be a single range taken as representative of a listing of numbers, typically the total of the numbers divided by what numbers are within the list.

Main Body:

Books owned by Michael = 97

Average book owned by people = sum of all books / total people

avg. book owned = (225+142+29+103+61)/5

avg book owned by people = 560/5

= 112

Hence books owned by avg people is 112.

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The probability that is lower than 63 grams is 95%

Given that,

X~ Normal ( 67 grams, 5 grams ) , n = 5

=> E( x ) = 67 grams , Var( X ) =5 grams

* ~ Normal ( 67 grams , 1 grams )

We know that, if X~ Normal ( E ( x) = a . Val (x ) = b)

Then X~ Normal ( E(x) = a, √Var (x)/n = √b/n)

E(x) = a, Var (x)/ = √b/n

Probability that sample mean X , is blw 65 and 70 grams

P ( 65< X<70 ) =

= P(Z<1. 76 ) - P(Z≤-2.65)

Using standard normal table values represent area

left of the Z- score, we get

= 0.96080 - 0.00402

= 0 .95678

=0.9568

So,The probability that is lower than 63 grams is 95%

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Yes, AAA Theorem is a valid theorem for congruency when used along with two more theorems to prove congruency of triangles.

AAA (Angle-Angle-Angle) Theorem states two corresponding angles of a triangle are equal to that of the second triangle if and only if the two corresponding sides containing the angles are proportional. Then the triangles are similar.

The AAA Theorem is valid for congruency of triangles if it follows the above given conditions along with the fact that two more theorems such as SSS or AAS.

As congruency in triangles is the condition in which both the given triangles are proven to be of the same shape and size.

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The probability that she succeeds at fewer than 7 free-throws is 0.19

How to determine the probability?

The table of values is given as

5: 0 successes

6: 5 successes

7: 8 successes

8: 6 successes

9: 6 successes

10: 1 success

From the above readings, the throws less than 7 are

5: 0 successes

6: 5 successes

So, the number of success is

Success = 0 + 5

Success = 5

The probability is then calculated as

P = Success/Total

So, we have

P = 5/(0 + 5 + 8 + 6 + 6 + 1)

Evaluate

P = 5/26

So, we have

P = 0.19

Hence, the probability is 0.19

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