A variable needs to be eliminated to solve the system of equations below. Choose the
correct first step
— 5х + бу = 41
7x - 6y = -43

(a) The expected number of good quality bottles in this batch is 7.2

(b) The probability that the process is running well, given that the second bottle is not of good quality, is 0.5

(c) The probability that the 5th bottle is of good quality is 0.84

(a) To find the expected number of good quality bottles in the batch of 10 bottles, we can multiply the probability of each bottle being of good quality by the total number of bottles.

Given that the probability of a bottle being of good quality is 90% if the process is running well and 60% if the process is not running well, and the probability that the process is running well is 80%, we can calculate the expected number of good quality bottles as follows:

Expected number of good quality bottles = (Probability of running well) × (Probability of good quality | running well) × (Total number of bottles)

= (0.8) × (0.9) × (10)

= 7.2

Therefore, the expected number of good quality bottles in this batch is 7.2.

(b) Given that the first 4 bottles tested are all of good quality except for the second bottle, we need to find the probability that the process is running well.

Let A be the event that the process is running well, and B be the event that the second bottle is not of good quality.

Using Bayes' theorem, we can calculate the probability:

P(A | B) = (P(B | A) × P(A)) / P(B)

P(B | A) is the probability that the second bottle is not of good quality given that the process is running well, which is 1 - 0.9 = 0.1.

P(A) is the probability that the process is running well, which is given as 0.8.

P(B) is the probability that the second bottle is not of good quality. To calculate this, we need to consider the cases where the process is running well and not running well.

P(B) = P(B | A) × P(A) + P(B | not A) × P(not A)

= 0.1 × 0.8 + 0.4 × 0.2

= 0.08 + 0.08

= 0.16

Now, we can calculate the probability using Bayes' theorem:

P(A | B) = (0.1 × 0.8) / 0.16

= 0.5

Therefore, the probability that the process is running well, given that the second bottle is not of good quality, is 0.5.

(c) Now, if the 5th bottle is tested as well, we want to find the probability that this bottle is of good quality.

Let C be the event that the 5th bottle is of good quality.

Using the law of total probability, we can calculate the probability:

P(C) = P(C | A) × P(A) + P(C | not A) × P(not A)

= 0.9 × 0.8 + 0.6 × 0.2

= 0.72 + 0.12

= 0.84

Therefore, the probability that the 5th bottle is of good quality is 0.84.

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The probability that none of the packs are defective is 0.478. The probability that five packs are defective is 0.000455.The probability that all the packs are defective is 2.8243e-14.The probability that 7 or more packs are defective is 0.00416 (approx).

A shipment of 10,000 Karak tea packs is produced by the company Chai Karak. If 7% of the packs are defective, what is the probability that: none of the packs are defective, five packs are defective, all the packs are defective, and seven or more packs are defective?  The number of trials, n, is 10 and the probability of a defective tea pack is 0.07. Therefore, the number of successful trials, X, follows a binomial distribution. Formula for binomial distribution: P(X = k) = nCk × pk × (1 − p)n−kWhere nCk = number of combinations of n things taken k at a time = n! / (k! (n-k)!)a. The probability that none of the packs are defective P(X = 0):P(X = 0) = nC0 * p0 * (1-p)n-0= 10C0 * 0.07^0 * (1-0.07)^10= 1 * 1 * 0.478= 0.478Therefore, the probability that none of the packs are defective is 0.478.

The probability that 5 packs are defective:P(X = 5) = nC5 * p^5 * (1-p)n-5= 10C5 * 0.07^5 * (1-0.07)^5= 252 * 0.0000028 * 0.649= 0.000455Therefore, the probability that five packs are defective is 0.000455.

The probability that all the packs are defective:P(X = 10) = nC10 * p^10 * (1-p)n-10= 10C10 * 0.07^10 * (1-0.07)^0= 1 * 2.8243e-14 * 1= 2.8243e-14Therefore, the probability that all the packs are defective is 2.8243e-14.

The probability that 7 or more packs are defective: P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)P(X = 7) = nC7 * p^7 * (1-p)n-7= 10C7 * 0.07^7 * (1-0.07)^3= 120 * 0.0000953677 * 0.657= 0.00416P(X = 8) = nC8 * p^8 * (1-p)n-8= 10C8 * 0.07^8 * (1-0.07)^2= 45 * 0.0000024969 * 0.859= 0.000011P(X = 9) = nC9 * p^9 * (1-p)n-9= 10C9 * 0.07^9 * (1-0.07)^1= 10 * 0.00000005 * 0.93= 4.65e-7P(X = 10) = nC10 * p^10 * (1-p)n-10= 10C10 * 0.07^10 * (1-0.07)^0= 1 * 2.8243e-14 * 1= 2.8243e-14P(X ≥ 7) = 0.00416 + 0.000011 + 4.65e-7 + 2.8243e-14= 0.00416Therefore, the probability that 7 or more packs are defective is 0.00416 (approx).

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The discount amount is $331.80. The price after discount is $616.20. The sales tax amount is $56.63. The final price is $672.83.

A TV originally priced at $948 is on sale for 35% off. We are to find the discount amount and the price after discount.

The original price of the TV = $948

The percentage discount = 35%.

Let X be the price after discount.

We can find X as follows:

Discount = 35% of original price

= 35% of 948= (35/100) × 948= $331.80

Price after discount (X) = Original price - Discount

= $948 - $331.80= $616.20

Therefore, the price after discount is $616.20.

Now we are to find the tax amount and the final price after including the discount and sales tax.

The sales tax is 9.2%.

We can find the tax amount as follows:

Tax amount = 9.2% of price after discount

= 9.2% of $616.20= (9.2/100) × 616.20= $56.63

Now, the final price after including the discount and sales tax = Price after discount + Tax amount

= $616.20 + $56.63= $672.83

Therefore, the final price after including the discount and sales tax is $672.83.

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Answer:

48

Step-by-step explanation:

To find the range, you have to subtract the biggest/largest number by the smallest number:

108 - 60 = 48

The range is 48.

Answer:

a=1

Step-by-step explanation:

Answer:

There should be 18 blue marbles in the representative sample.

Step-by-step explanation:

Given that a bag contains 9 white and 54 blue marbles, if a representative sample contains 3 white marbles, to determine how many blue marbles would you expect to contain the following calculation must be performed:

9 = 3

54 = X

54 x 3/9 = X

162/9 = X

18 = X

Therefore, there should be 18 blue marbles in the representative sample.

Answer:

43 feet would be the correct answer.

Step-by-step explanation:

The value of x that satisfies the equation (81)(27)^(2x-5) - 9^(3-4x) = 0 is x = 17/14.

To find the value of x in the equation (81)(27)^(2x-5) - 9^(3-4x) = 0, we can use the properties of exponents and logarithms to simplify and solve the equation. By equating the bases and exponents on both sides, we can determine the value of x.

We start by simplifying the equation. Applying the exponent properties, we have (3^4)(3^3)^(2x-5) - (3^2)^(3-4x) = 0.

Simplifying further, we get (3^(4 + 3(2x-5))) - (3^(2(3-4x))) = 0.

Using the property (a^b)^c = a^(b*c), we can rewrite the equation as 3^(4 + 6x - 15) - 3^(6 - 8x) = 0.

Combining like terms, we have 3^(6x - 11) - 3^(6 - 8x) = 0.

To equate the bases and exponents, we set 6x - 11 = 6 - 8x.

Simplifying the equation, we get 14x = 17.

Dividing both sides by 14, we find that x = 17/14.

Therefore, the value of x that satisfies the equation (81)(27)^(2x-5) - 9^(3-4x) = 0 is x = 17/14.

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Answer:

50% increase

Step-by-step explanation:

The percentage change in volume from cylinder A to cylinder B is 50% volume increased by 50% option (C) is correct.

In geometry, it is defined as the three-dimensional shape having two circular shapes at a distance called the height of the cylinder.

We know the formula for the volume of the cylinder is given by:

[tex]\rm V = \pi r^2h[/tex]

We have r = 10 inches and h = 5 inches

[tex]\rm V = \pi \times10^2\times5[/tex]

V = 500π cubic inches

Percent change in volume from cylinder A to cylinder B:

[tex]=\rm \frac{750\pi-500\pi}{500\pi} \times100[/tex]

= 50%

Thus, the percentage change in volume from cylinder A to cylinder B is 50% volume increased by 50% option (C) is correct.

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Answer:

4

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

2/3 x 6 is 4 because when you times the 2 in 2/3 u get 12 and 12/3 is the same as 4

Answer:

The correct answer is (A).

Step-by-step explanation:

Because the data came from 3 groups in a randomized experiment, the appropriate test is a chi-square test for homogeneity.

Answer:

Yes

Step-by-step explanation:

Hello There!

Remember the sum of the two shorter sides has to be greater than the largest side in order for it to be a triangle

11+12 must be greater than 13

11+12=23

23>13 therefore the side lengths can form a triangle

Answer:

Kindly check explanation

Step-by-step explanation:

Given that, the number of people who liked Mindy's website is given by :

270(1.65)^t ; t = number of weeks after a review

Interpreting the equation :

The equation represents an exponential growth function :

y = a(1 + r)^t

a = initial amount ; r = growth rate ; t = period

Comparing the function with the equation given ;

270 = a = number of people who liked the website before the review

1 + r = 1.65

r = 1.65 - 1

r = 0.65 ; 0.65 * 100% = 65%

Site grew at a rate of 65% per week after review

Ther predicted number of likes 2 weeks after the review :

y = 270(1.65)^2

y = 270 * 2.7225

y = 735.075

Hence, number of likes after 2 weeks = 735

Answer:3.46

Step-by-step explanation:

Answer:

11 upon 4 is your 6 upon 4 answer

Answer:

3 to 4

Step-by-step explanation:

12÷4=3

16÷4=4

so that makes it 3 to 4

The answer is A. 2pi/3 radians.

Answer:

First Option

Explanation:

A)  2pi/3 radians

Hope this helps :)

Answer:

$1,462.50

Step-by-step explanation:

We are to find the total amount after 4 years

Principal => $1,250

Rate => 4.25%

The formula to find the total amount =

A = P(1 + rt)

First, converting R percent to r a decimal

r = R/100 = 4.25%/100 = 0.0425 per year.

Solving our equation:

A = 1250(1 + (0.0425 × 4)) = 1462.5

A = $1,462.50

The total amount accrued, principal plus interest, from simple interest on a principal of $1,250.00 at a rate of 4.25% per year for 4 years is $1,462.50.

Answer:

-4.91 feet

Step-by-step explanation:

Mean is a measure of central tendency. It is the average of a set of numbers. It is determined by adding the numbers together and dividing it by the total number

Mean = Total change in altitude / total time

378 / 77 = 4.91

Given an adjacency matrix, we need to find the graph which has the following adjacency matrix:[2 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0]We know that an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.The diagonal elements of the matrix are always zero since self-loops are not allowed in a simple graph.The (i, j)-entry of the matrix is 1 if the ith vertex and jth vertex are adjacent, and 0 otherwise. Hence, for the given matrix, there is an edge between vertex 1 and vertex 2 and an edge between vertex 1 and vertex 3. Similarly, there are other edges between vertices.Here is the graph which has the given adjacency matrix:Graph with adjacency matrixFrom the graph above, we can label the vertices A, B, C, D in any order. Let's say we label the vertices as A = vertex 1, B = vertex 2, C = vertex 3, and D = vertex 4.Now, we need to find the number of different paths of length 10 from vertex A to vertex D.The number of paths of length n from vertex i to vertex j in a graph is given by the (i, j)-entry in the n-th power of the adjacency matrix. Here, we need to find the number of paths of length 10 from vertex 1 to vertex 4. So, we need to compute the 10th power of the given adjacency matrix:[2 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0] ^ 10We can use a calculator or software like MATLAB to compute the matrix power. After computing, we get that the (1, 4)-entry of the 10th power of the matrix is 8. Therefore, there are 8 different paths of length 10 from vertex A to vertex D.

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Answer:23/8Step-by-step explanation:(LCM of 4,8 and 6 = 24 )

3×6/4×6 + 7×3/8×3 + 5×4/6×4 28+21+20/24

59/24

I hope you got it❤️❤️❤️And thanks your patience ❤️.SORRY

Answer:

True

Step-by-step explanation:

With Sin and Cos, the rule is that the opposites are the same:

SinA=CosB

CosB=SinA

In this problem, it is showing SinA=cosB, so it is the same, it is true.

Hope this helps!

The value of the test statistic in the scenario given above is 3.487

The test statistic can be calculated as follows:

t = (mean - hypothesized mean) / (standard deviation / √(sample size))

mean = 98.7

hypothesized mean = 98.5

standard deviation = 0.5

sample size = 76

Plugging these values into the formula, we get the following test statistic:

t = (98.7 - 98.5) / (0.5 / √(76)) = 3.487

Therefore, the value of the test statistic in the above scenario is 3.487

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The probability of getting a negative result is 0.175 or 17.5%.

To calculate the probability of getting a negative result, we need to consider the sensitivity and specificity of the diagnostic test.

Given that the prevalence of the condition is 12.5%, we can assume that 12.5% of the population has the condition, and the remaining 87.5% does not.

The sensitivity of the test is 80%, which means that it correctly identifies 80% of the individuals with the condition as positive.

The specificity of the test is 95%, which means that it correctly identifies 95% of the individuals without the condition as negative.

To calculate the probability of getting a negative result, we need to consider both the true negative rate (1 - sensitivity) and the proportion of individuals without the condition (1 - prevalence).

Probability of getting a negative result = (1 - sensitivity) × (1 - prevalence)

= (1 - 0.80) × (1 - 0.125)

= 0.20 * 0.875

= 0.175

Therefore, the probability of getting a negative result is 0.175 or 17.5%.

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(i) Estimate X given that Y = 0.5 and Z = -3 using an unbiased minimum variance estimator.

Estimate of X given that Y = 0.5 and Z = -3 can be obtained by applying the conditional expectation formula, E[X|Y=y, Z=z], where y=0.5 and z=-3.E[X|Y=y, Z=z] = E[X] + Cov[X,Y]/Var[Y] * (Y - E[Y]) + Co v[X,Z]/Var[Z] * (Z - E[Z])E[X|Y=0.5, Z=-3] = 0 + (0/10) * (0.5 - 1) + (1/2) * (-3 - 2) = -2, which is the unbiased minimum variance estimator.(ii) Determine the variance of the above estimator.

The variance of the unbiased minimum variance estimator is given by Var[X|Y=y, Z=z] = Var[X] - Cov[X,Y]^2/Var[Y] - Cov[X,Z]^2/Var[Z] + 2Cov[X,Y]Cov[X,Z]/(Var[Y]*Var[Z])Var[X|Y=0.5, Z=-3] = 10 - 0^2/10 - 1^2/2 + 2(0)(1)/(10*2) = 9.75 (b)

Intelli Moto is car manufacturer that produces vehicles equipped with a fault detection system that uses information from various sensors to inform the driver about possible faults in the braking system. The system diagnoses faults correctly with probability 99%, but gives false alarms with probability 2%. It is known that such faults occur with probability 0.05%. If the system diagnoses a fault, what is the probability a fault has actually occured?

The probability of a fault actually occurring is P(Fault) = 0.05%, which is the prior probability of a fault.

The probability of a correct diagnosis is P(Diagnosis | Fault) = 99%, which is the probability of a positive test result given that a fault has actually occurred.

The probability of a false alarm is P(Diagnosis | No Fault) = 2%, which is the probability of a positive test result given that no fault has actually occurred.

The probability of a positive test result isP(Diagnosis) = P(Fault)*P(Diagnosis | Fault) + P(No Fault)*P(Diagnosis | No Fault)= 0.05% * 99% + 99.95% * 2% = 2.039%.The probability of a fault given a positive test result can be obtained by Bayes' theorem,P(Fault | Diagnosis) = P(Diagnosis | Fault)*P(Fault)/P(Diagnosis)= 99% * 0.05% / 2.039% = 2.43%, which is the probability a fault has actually occurred given that the system diagnoses a fault.

The probability that a fault has actually occurred given that the system diagnoses a fault is 2.42%.

(i) To estimate X given that Y = 0.5 and Z = -3 using an unbiased minimum variance estimator, we need to determine the distribution of X | Y = 0.5, Z = -3 and use the formula for conditional expectation of a jointly normally distributed random variable. The distribution of X | Y = 0.5, Z = -3 is also normal since it is a conditional distribution of a jointly normally distributed random variable. To find the mean of the distribution, we use the formula for conditional expectation:
[tex]E[X | Y = 0.5, Z = -3] = E[X] + Cov[X, Y | Z = -3] (Y - E[Y | Z = -3]) / Var[Y | Z = -3] + Cov[X, Z | Y = 0.5] (Z - E[Z | Y = 0.5]) / Var[Z | Y = 0.5][/tex]

where Cov[X, Y | Z = -3] is the conditional covariance of X and Y given Z = -3,

E[Y | Z = -3] is the conditional mean of Y given Z = -3,

Var[Y | Z = -3] is the conditional variance of Y given Z = -3,

Cov[X, Z | Y = 0.5] is the conditional covariance of X and Z given Y = 0.5,

and E[Z | Y = 0.5] and Var[Z | Y = 0.5] are the conditional mean and variance of Z given Y = 0.5 respectively.

We are given that

E[X] = 0, E[Y] = 1, E[Z] = 2,

Var[X] = 10, Var[Y] = 2, Var[Z] = 1,

and Cov[X, Y] = Cov[X, Z] = Cov[Y, Z] = 0.

Also, Y = 0.5 and Z = -3.

Hence, we have:

[tex]Cov[X, Y | Z = -3] = Cov[X, Y] / Var[Z] = 0[/tex],

[tex]E[Y | Z = -3] = E[Y] =[/tex]1,

[tex]Var[Y | Z = -3] = Var[Y] = 2[/tex],

[tex]Cov[X, Z | Y = 0.5] = Cov[X, Z] / Var[Y] = 0[/tex].

The conditional mean of Z given Y = 0.5 is given by

[tex]E[Z | Y = 0.5] = E[Z] + Cov[Y, Z] (Y - E[Y]) / Var[Y] = 2 + 0.5 (0 - 1) / 2 = 1.5.[/tex]

The conditional variance of Z given Y = 0.5 is given by

[tex]Var[Z | Y = 0.5] = Var[Z] - Cov[Y, Z]^2 / Var[Y] = 1 - 0^2 / 2 = 1[/tex].

Hence, the mean of the distribution of X | Y = 0.5, Z = -3 is:
[tex]E[X | Y = 0.5, Z = -3] = 0 + 0 (0.5 - 1) / 2 + 0 (-3 - 1.5) / 1 = -0.75[/tex]

To find the variance of the unbiased minimum variance estimator, we use the formula for conditional variance of a jointly normally distributed random variable:
[tex]Var[X | Y = 0.5, Z = -3] = Var[X] - Cov[X, Y | Z = -3]^2 / Var[Y | Z = -3] - Cov[X, Z | Y = 0.5]^2 / Var[Z | Y = 0.5][/tex]

where Var[X], Cov[X, Y | Z = -3], and Cov[X, Z | Y = 0.5] are given above,

and Var[Y | Z = -3] and Var[Z | Y = 0.5] are calculated as follows:
[tex]Var[Y | Z = -3] = Var[Y] - Cov[X, Y]^2 / Var[Z] = 2 - 0^2 / 1 = 2Var[Z | Y = 0.5] = Var[Z] - Cov[Y, Z]^2 / Var[Y] = 1 - 0^2 / 2 = 1[/tex]

Hence, we have:
[tex]Var[X | Y = 0.5, Z = -3] = 10 - 0^2 / 2 - 0^2 / 1 = 10[/tex]

(ii) The variance of the unbiased minimum variance estimator is Var[X | Y = 0.5, Z = -3] = 10.

(b) Let A denote the event that a fault has actually occurred, D denote the event that the system diagnoses a fault,

P(A) = 0.05%, P(D | A) = 99%, and P(D | A') = 2%, where A' is the complement of A.

We need to find P(A | D), the probability that a fault has actually occurred given that the system diagnoses a fault.

By Bayes' theorem, we have:
[tex]P(A | D) = P(D | A) P(A) / P(D)[/tex]

where P(D) is the total probability of the system diagnosing a fault, which is:
[tex]P(D) = P(D | A) P(A) + P(D | A') P(A') = 0.99 (0.0005) + 0.02 (1 - 0.0005) = 0.0205[/tex]

Hence, we have:
[tex]P(A | D) = 0.99 (0.0005) / 0.0205 = 0.0242[/tex] or 2.42%

Therefore, the probability that a fault has actually occurred given that the system diagnoses a fault is 2.42%.

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