Answer:
5.333 hours
Step-by-step explanation:
We know
4 Pipes fill a tank in 16 hours.
How long will it take to fill the tank if 12 pipes of the same dimensions are used?
We Take
16 x 1/3 = 5.333 hours
So, it takes about 5.333 hours to fill the tank.
Anna wants to make 30 mL of a 60 percent salt solution by mixing togethera 72 percent salt solution and a 54 percent salt solution. How much of each solution should dhe use
Anna should use 10 mL of the 72% salt solution and 20 mL of the 54% salt solution to make 30 mL of a 60% salt solution
Let's assume that Anna will use x mL of the 72% salt solution, and therefore she will use (30 - x) mL of the 54% salt solution (since the total volume is 30 mL).
To find out how much of each solution Anna should use, we can set up an equation based on the amount of salt in each solution.
The amount of salt in x mL of 72% salt solution is
= 0.72x
The amount of salt in (30 - x) mL of 54% salt solution is
= 0.54(30 - x)
To make a 60% salt solution, the total amount of salt in the final solution should be
0.6(30) = 18
So we can set up an equation
0.72x + 0.54(30 - x) = 1
Simplifying the equation
0.72x + 16.2 - 0.54x = 18
0.18x = 1.8
x = 10 ml
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Suppose E and F are two events, with the following probability table F F’
E 0.1 0.3 E' 0.2 0.4 a) Compute P(EF). b) Are E and F independent? Explain. c) Are E and F mutually exclusive? Explain.
a) With the following probability table F F, Let’s apply the formula for the intersection of events to solve the first part of the problem.
P(EF) = P(E) x P(F|E).We know that P(E) = 0.1 and that P(F|E) = 0.3. Therefore,P(EF) = P(E) x P(F|E) = 0.1 x 0.3 = 0.03.b) Two events E and F are independent if and only if their intersection is equal to the product of their individual probabilities.
P(EF) = P(E) x P(F) if and only if E and F are independent. We know that P(E) = 0.1 and that P(F) = 0.1 + 0.3 = 0.4. Therefore, P(EF) = 0.03, which is different from 0.1 x 0.4 = 0.04.
Since P(EF) is different from P(E) x P(F), it means that E and F are not independent.c) Two events E and F are mutually exclusive if and only if their intersection is the null set.P(EF) = ∅ if and only if E and F are mutually exclusive. We know that P(EF) = 0.03, which is not equal to the null set. Therefore, E and F are not mutually exclusive.
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If the GM between √2 and 2√2 is a find the value of a.
Answer:
If the GM between √2 and 2√2 is a find the value of a.
Step-by-step explanation:
To find the geometric mean between two numbers, we simply take the square root of their product.
In this case, we want to find the geometric mean between √2 and 2√2.
Their product is:
√2 * 2√2 = 2√4 = 2*2 = 4
So, the geometric mean between √2 and 2√2 is the square root of 4, which is:
√4 = 2
Therefore, the value of a is 2.
Which points satisfy both inequalities?
The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.
What is inequality?An inequality is a mathematical statement that cοmpares twο values οr expressiοns and indicates whether they are equal οr nοt, οr which οne is greater οr smaller.
Since the shading is nοt included, we will need tο use the lines themselves tο determine the cοrrect regiοn οf the cοοrdinate plane.
The first inequality y > (3/2)x - 5 has a slοpe οf 3/2 and a y-intercept οf -5. This means the line will have a pοsitive slοpe and will be lοcated belοw the pοint (0,-5).
The secοnd inequality y < (-1/6)x - 6 has a negative slοpe οf -1/6 and a y-intercept οf -6. This means the line will have a negative slοpe and will be lοcated abοve the pοint (0,-6).
Tο find the pοint that satisfies BOTH inequalities, we need tο lοοk fοr the regiοn οf the cοοrdinate plane that is belοw the line y = (3/2)x - 5 AND abοve the line y = (-1/6)x - 6. This regiοn is the triangular-shaped area that is bοunded by the twο lines and the x-axis.
The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.
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Aaron sampled 101 students and calculated an average of 6.5 hours of sleep each night with a standard deviation of 2.14. Using a 96% confidence level, he also found that t* = 2.081.confidence intervat = x±s/√n A 96% confidence interval calculates that the average number of hours of sleep for working college students is between __________.
The average number of hours of sleep for working college students is between 6.28 and 6.72 hours of sleep each night
According to the given data,
Sample size n = 101
Sample mean x = 6.5
Standard deviation s = 2.14
Level of confidence C = 96%
Using a 96% confidence level, the value of t* for 100 degrees of freedom is 2.081, as given in the question.
Now, the formula for the confidence interval is:x ± (t* × s/√n)Here, x = 6.5, s = 2.14, n = 101, and t* = 2.081
Substituting the values in the above formula, we get:
Lower limit = x - (t* × s/√n) = 6.5 - (2.081 × 2.14/√101) = 6.28
Upper limit = x + (t* × s/√n) = 6.5 + (2.081 × 2.14/√101) = 6.72
Therefore, the 96% confidence interval for the average number of hours of sleep for working college students is between 6.28 and 6.72 hours of sleep each night.
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The exponential 12 (3) 2x-12 has been converted to 12(k)*-6, what is the value of k?
Answer:
The solution set is (13,− 32). A quadratic equation of the form x 2= k can be solved by factoring with the following sequence of equivalent equations.
Step-by-step explanation:
Which of the following are true statements? Check all that apply. A. F(x)= 2 square x has the same domain and range as f(x)= square x. B. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2. C. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it horizontally by a factor of 1/2. D. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.
The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2.
The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.
Thus, Option B and Option D are correct.
What is function?A function is a relationship or expression involving one or more variables. It has a set of input and outputs.
A. F(x)= 2 square x has the same domain and range as f(x)= square x.
B. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2.
D. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.
Option A is false because multiplying the function by 2 will change the range of the function to include all non-negative real numbers (since the square of any number is non-negative).
Option B is true because multiplying the function by 2 will vertically shrink the graph by a factor of 1/2 (since the output values will be half the size of the original function).
Option C is false because multiplying the function by 2 will not affect the horizontal scale of the graph.
Option D is true because multiplying the function by 2 will vertically stretch the graph by a factor of 2 (since the output values will be twice the size of the original function).
Therefore, Option B and Option D are correct.
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consider using a z test to test h0: p 5 .6. determine the p-value in each of the following situations. a. ha:p..6,z51.47 b. ha:p,.6,z522.70 c. ha:p?.6,z522.70 d. ha:p,.6,z5.25
a) P-value = P(z<1.47) = 0.9292.
b) P-value = P(z>2.70) = 0.0036.
c) P-value = 2 × P(z>2.70) = 0.0072.
d) P-value = P(z>2.5) = 0.0062.
Z-test is a statistical test for the null hypothesis, which refers to the population mean, where the population standard deviation is known. P-value represents the probability value for any hypothesis, where a small p-value indicates that the null hypothesis is less accurate.
P-value, for the given values of z-test is calculated as follows: a) For ha: p < .6, z=1.47The p-value for this hypothesis test is calculated as follows: P-value = P(z<1.47) = 0.9292. Therefore, the P-value is 0.9292. b) For ha: p > .6, z=2.70The p-value for this hypothesis test is calculated as follows.
P-value = P(z>2.70) = 0.0036. Therefore, the P-value is 0.0036.c) For ha: p ≠ .6, z=2.70The p-value for this hypothesis test is calculated as follows: P-value = 2 × P(z>2.70) = 0.0072.
Therefore, the P-value is 0.0072.d) For ha: p > .6, z=2.5The p-value for this hypothesis test is calculated as follows: P-value = P(z>2.5) = 0.0062. Therefore, the P-value is 0.0062.
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Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order.
y dA, D is bounded by y = x − 6; x = y2
D
The value of the double integral using the easier order, ydA bounded by y = x − 6; x = y² is 125/12.
The double integral, indicated by ', is mostly used to calculate the surface area of a two-dimensional figure. By using double integration, we may quickly determine the area of a rectangular region. If we understand simple integration, we can easily tackle double integration difficulties. Hence, first and foremost, we will go over some fundamental integration guidelines.
Given, the double integral ∫∫yA and the region y = x-6 and x = y²
y = x-6
x = y²
y² = y +6
y² - y - 6 = 0
y² - 3y +2y - 6 = 0
(y-3) (y+2) = 0
y = 3 and y = -2
[tex]\int\int\limits_\triangle {y} \, dA\\ \\[/tex]
= [tex]\int\limits^3_2 {y(y+6-y^2)} \, dx \\\\\int\limits^3_2 {(y^2+6y-y^3)} \, dx \\\\(\frac{y^3}{3} + 3y^2-\frac{y^4}{4} )_-_2^3\\\\\frac{63}{4} -\frac{16}{3} \\\\\frac{125}{12}[/tex]
The value for the double integral is 125/12.
Integration is an important aspect of calculus, and there are many different forms of integrations, such as basic integration, double integration, and triple integration. We often utilise integral calculus to determine the area and volume on a very big scale that simple formulae or calculations cannot.
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according to a census, 3.3% of all births in a country are twins. if there are 2,500 births in one month, calculate the probability that more than 90 births in one month would result in twins. use a ti-83, ti-83 plus, or ti-84 calculator to find the probability. round your answer to four decimal places. provide your answer below:
According to a census, 3.3% of all births in a country are twins. In a month, there are 2,500 births. The census reports that 3.3% of all births result in twins, and the probability of having more than 90 twins in a month is "0.4351."
We will solve this problem using the binomial distribution formula, which is as follows:P (X > 90) = 1 - P (X ≤ 90)where P represents the probability, X represents the number of twins born in a month, and X is a binomial random variable with a sample size of n = 2,500 and a probability of success (having twins) of p = 0.033. Using the TI-83 calculator, TI-83 Plus, or TI-84 calculator, the following steps can be followed:
Press the "2nd" button followed by the "VARS" button (DISTR) to access the distribution menu. Scroll down and select "binomcdf (" from the list of options (use the arrow keys to navigate). The binomcdf ( menu will appear on the screen. The first number in the parentheses is the number of trials, n, and the second number is the probability of success, p. We want to find the probability of having more than 90 twins, so we need to use the "compliment" option. Therefore, we will subtract the probability of having 90 twins or less from 1 (using the "1 -" key). Type in "binomcdf (2500,0.033,90)" and press the "ENTER" button on your calculator.
This will give you the probability of having 90 twins or fewer in a month. Subtract this value from 1 to obtain the probability of having more than 90 twins in a month, which is the answer to our question. P(X>90) = 1 - binomcdf (2500,0.033,90)P(X>90) = 1 - 0.5649P(X>90) = 0.4351Therefore, the probability of having more than 90 twins in a month is 0.4351.
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Identify the fallacies of relevance committed by the following arguments, giving a brief explanation for your answer. If no fallacy is committed, write "no fallacy". Surely you welcome the opportunity to join our protective organization. Think of all the money you will lose from broken windows, overturned trucks, and damaged merchandise in the event of your not joining.
There are no fallacies of relevance committed by the given argument.
The following arguments commits fallacy: argumentum ad baculum. Argumentum ad baculum is a Latin phrase which means argument from a stick or appeal to force. It is a type of logical fallacy in which someone tries to persuade another person by using threats of force or coercion rather than using evidence or reasoning.
The above statement is an example of the argumentum ad baculum fallacy as it tries to use fear to convince people to join their protective organization. They are using the threat of potential losses to convince people to join. It is a manipulative strategy that attempts to scare people into joining by threatening the safety of their business.No fallacy is committed. There are no fallacies of relevance committed by the given argument.
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A satellite TV company offers two plans. One plan costs $115 plus $30 per month. The other plan costs $60 per month. How many months must Alfia have the plan in order for the first plan to be the better buy?
Solve the following formula for t
S=12(V0+V1)t
Answer:
[tex]{ \rm{s = 12( v_{0} + v_{1} )t}} \\ \\{ \boxed { \rm{t = \frac{s}{12(v_{0} + v_{1})} \: \: }}}[/tex]
a bin can hold 28 pounds. each toy car weighs 7 ounces. how many toy cars can the bin hold? (2 points) 64 toy cars 72 toy cars 88 toy cars 92 toy cars
A bin can hold 28 pounds. each toy car weighs 7 ounces., so the bin can hold 64 toy cars.
How to determine the number of toy carsTo determine the number of toy cars the bin can hold, we must first convert the weight limit of the bin and the weight of the toy cars to a uniform unit of measure.
We'll then divide the weight limit of the bin by the weight of one toy car. After that, we'll multiply the resulting value by the number of ounces in one pound (16).
Here's how to solve the problem:
1 pound = 16 ounces
Therefore, a bin that can hold 28 pounds can hold:28 × 16 = 448 Ounces
The weight of one toy car is 7 ounces.
Divide the weight limit of the bin (448 ounces) by the weight of one toy car (7 ounces):
448 ÷ 7 = 64
Therefore, the bin can hold 64 toy cars.
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Find the outer perimeter.
6 ft
4 ft
15 ft
10 ft
P = [?] ft
Round to the nearest
hundredth.
Answer:
P= 40 ft
Step-by-step explanation:
Perimeter is the sum of all the lengths
So,
Perimeter= 6+4+15+10ft
= 35ft
Nearest ten can be 40ft or 30ft
If you succeed In understanding then kindly mark my answer the brainliest. Thank you :)
Q4.
The diagram shows a regular hexagon OABCDE.
a
E
OA = a
AB = b
M is the midpoint of OE.
N is the midpoint of AB.
(a) Find MN in terms of a and/or b.
b
B
D
Diagram NOT
accurately drawn
By answering the presented question, we may conclude that So, the Pythagorean theorem length of MN is expressed in terms of a and b.
What is Pythagorean theorem?Its Pythagorean theorem is just a fundamental mathematical principle that explains the connection between the sides of a triangle that is right. It asserts that the sum of the squares of both the widths of the other two sides is a square of both the width of the hypotenuse (the side facing the perfect angle) the side opposite the right angle). The mathematical mathematics is as follows: c2 = a2 + b2 At which "c" indicates the length of the right triangle and "a" and "b" reflect the extents of the additional two sides, started referring to as the legs.
Because M is the midpoint of OE and N is the midpoint of AB, we can draw a line segment connecting M and N that is parallel to OB and AE and perpendicular to AB.
the Pythagorean theorem
[tex]OE² = OX² + XE²OE²[/tex]
[tex](a + b/2)² + (2a - b/√3)²OE² = 7a²/4 + 3ab/2 + b²/4AN²[/tex]
[tex]AE² + EN²AN² = (2a√3)² + (b/2)²AN²[/tex]
[tex]12a² + b²/4MN² = AN² + AM²MN² \\\\ 12a² + b²/4 + (7a²/4 + 3ab/2 + b²/4)MN²\\\\19a²/2 + 3ab/2 + b²/2MN = √(19a²/2 + 3ab/2 + b²/2)[/tex]
So, the length of MN is expressed in terms of a and b.
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I need help please show your work
Answer:
The 2nd equation is false.
Step-by-step explanation:
You don't even have to solve. DE is not 58, it's 40.
The 2nd equation is false.
If P = 2y² + 4xy + 4
Q = − 3y² + 7 - 3xy
R=- 3xy + 8
Find P+Q=R.
Answer:
P = [tex]2y^{2}[/tex] + 4xy +4
Q = [tex]-3y^{2}[/tex] + 7 -3xy
R = -3xy +8
Step-by-step explanation:
for h(x) = 4x-1, find h(0) and h(2)
Answer:
- 1 and 7
Step-by-step explanation:
to find h(0) substitute x = 0 into h(x)
h(0) = 4(0) - 1 = 0 - 1 = - 1
to find h2) substitute x = 2 into h(x)
h(2) = 4(2) - 1 = 8 - 1 = 7
I cant figure it out
4x - 4/4x² + x is the value of linear equation.
What in mathematics is a linear equation?
A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.
Equations with power 1 variables are known as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.
28x³ - 28x²/28x⁴ + 7x³
= 28x²( x - 1 )/7x³( 4x + 1)
= 4( x - 1)/x( 4x + 1)
= 4x - 4/4x² + x
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The food service manager conducted a random survey of 200 students to determine their preference for new lunch menu items. There are 1,500 students in the school. Select all the manager’s predictions that are supported by the data
There are several predictions that the food service manager may make based on the data from the survey of 200 students regarding their preference for new lunch menu items. Let's examine some of these predictions and see if they are supported by the data.
The majority of students will like the new menu items.
The food service manager may predict that the majority of students in the school will like the new menu items, based on the positive responses from the 200 surveyed students. However, it's important to note that the sample size of 200 is relatively small compared to the total student population of 1,500. Therefore, it's possible that the preferences of the 200 surveyed students may not be representative of the preferences of the entire student population. To make a more accurate prediction, the manager may need to conduct a larger survey or pilot program to test the new menu items with a larger group of students.
Certain menu items will be more popular than others.
Based on the survey data, the food service manager may be able to identify which new menu items are more popular among the surveyed students. For example, if a majority of students indicate that they would like to see more vegetarian options, the manager may predict that introducing more vegetarian menu items will be popular among the broader student population. However, it's important to keep in mind that the preferences of the 200 surveyed students may not be representative of the preferences of the entire student population, so the manager may need to conduct additional research or testing to confirm these predictions.
The introduction of new menu items will increase overall satisfaction with the school lunch program.
If the survey data shows that a significant number of students are excited about the new menu items, the food service manager may predict that introducing these items will increase overall satisfaction with the school lunch program. However, it's important to note that satisfaction is a complex concept that can be influenced by many factors beyond just the menu items, such as the quality of service, cleanliness of the cafeteria, and overall atmosphere. Therefore, the manager may need to consider these other factors when predicting the impact of the new menu items on overall satisfaction with the lunch program.
In summary, while the data from the survey of 200 students can provide valuable insights into student preferences for new lunch menu items, it's important to interpret these results with caution and consider additional factors that may influence the broader student population. Conducting further research or testing can help to confirm these predictions and make more accurate decisions about the school lunch program.
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In a candy factory, each bag of candy contains 300 pieces. The bag can be off by 10 pieces.
Write an absolute value inequality that displays the possible number of candy pieces that a bag contains.
Answer:
[tex] |x - 300| \leqslant 10[/tex]
solve the proportion 7/11=18/x+1
Solve the equation [tex]7/11=18/x+1[/tex] we find the solution is [tex]x = 27.2857[/tex]
What is a formula or equation?Your example is an equation since an equation is any statement with an equals sign. Equations are frequently utilized for mathematical equations since mathematicians like equal signs. A set of instructions for achieving a certain result is called an equation.
A formula is it an expression?A number, a constant, or a mix of numbers, variables, and operation symbols make up an expression. Two expressions joined by such an assignment operator form an equation.
we can cross-multiply,
[tex]7(x+1) = 11(18)[/tex]
Expanding the left side,
[tex]7x + 7 = 198[/tex]
Subtracting [tex]7[/tex] from both sides,
[tex]7x = 191[/tex]
Dividing both sides by [tex]7[/tex],
[tex]x = 191/7[/tex]
Therefore, the solution to the proportion is
[tex]x = 27.2857[/tex]
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If the midpoint of 2 sides of a triangle are connected with a segment then
The Midpoint is the middle- point of the line member. The midpoint connecting two sides of a triangle is resemblant to the third side and half as long.
The midpoint is the middle of the line member. It's equidistant from both endpoints and is the centroid of the member and endpoints. Cut a member in two.
The midpoint theorem states that a line member drawn from the midpoint of two sides of a triangle is resemblant to the third side and half the length of the third side of the triangle.
The mean theorem helps us find the missing values for the sides of triangles. Connects the sides of a triangle with a line member drawn from the midpoints of two sides of the triangle.
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what is the as surface area of the rectangular prism
Answer:
142 sq cm
Step-by-step explanation:
A= 2(lh + wh + lw)
2(7*3+5*3+7*5)
2(21+15+35)
2(71)
A= 142 sq cm
use the trapezoidal rule and simpson's rule to approximate the value of the definite integral for the given value of n. round your answer to four decimal places and compare the results with the exact value of the definite integral. 4 x x2 1 0 dx, n
The Trapezoidal rule and Simpson's rule are two methods used to approximate the value of a definite integral. The Trapezoidal rule approximates the integral by dividing the region between the lower and upper limits of the integral into n trapezoids, each with a width h. The approximate value of the integral is then calculated as the sum of the areas of the trapezoids. The Simpson's rule is similar, except the region is divided into n/2 trapezoids and then the integral is approximated using the weighted sum of the area of the trapezoids.
For the given integral 4 x x2 1 0 dx, with n = 200, the Trapezoidal rule and Simpson's rule approximate the integral to be 7.4528 and 7.4485 respectively, rounded to four decimal places. The exact value of the integral is 7.4527. The difference between the exact and approximate values is very small, thus indicating that both the Trapezoidal rule and Simpson's rule are accurate approximations.
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determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span. a, S = {(1, −1), (2, 1)} b, S = {(1, 1)} c, S = {(0, 2), (1, 4)}
a. S = {(1, -1), (2, 1)}Let's begin by calculating the determinant of the matrix composed of the vectors of S, and checking if it is equal to 0. Because the two vectors are not colinear, they should span R2.|1 -1||2 1| determinant is not 0, therefore S spans R2. No geometric description is required for this example.
b. S = {(1, 1)} The set S contains one vector. A set containing only one vector cannot span a plane because it only spans a line. Therefore, S does not span R2. Geometric description: S spans a line that passes through the origin (0, 0) and the point (1, 1).c. S = {(0, 2), (1, 4)} Let's again begin by calculating the determinant of the matrix composed of the vectors of S, and checking if it is equal to 0.|0 2||1 4| determinant is 0, thus S does not span R2. In this scenario, S only spans the line that contains both vectors, which is the line with the equation y = 2x.
Geometric description: S spans a line that passes through the origin (0, 0) and the point (1, 2).
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Student A can solve 75% of problems, student B can solve 70%. What is the probability that A or B can solve a problem chosen at random?
The probability that student A or B can solve a problem chosen at random is 0.95.
Probability is calculated by dividing the number of favourable outcomes by the number of possible outcomes.
Random: An event is referred to as random when it is not possible to predict it with certainty. The probability that either student A or B will be able to solve a problem chosen at random can be calculated as follows:
P(A or B) = P(A) + P(B) - P(A and B) where: P(A) = probability of A solving a problem = 0.75, P(B) = probability of B solving a problem = 0.7, P(A and B) = probability of both A and B solving a problem. Since A and B are independent, the probability of both solving the problem is:
P(A and B) = P(A) x P(B) = 0.75 x 0.7 = 0.525
Now, using the above formula: P(A or B) = P(A) + P(B) - P(A and B) = 0.75 + 0.7 - 0.525 = 0.925
Therefore, the probability that student A or B can solve a problem chosen at random is 0.95 (or 95%).
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if the circumference of the moon is 6783 miles what is its diameter in miles
Answer:
C = 21,309.4
Step-by-step explanation:
Diameter of moon is miles is,
d = 2159.8 miles
We have,
The circumference of the moon is, 6783 miles
Since, We know that,
the circumference of circle is,
C = 2πr
Substitute given values,
6783 miles = 2 × 3.14 × r
6783 = 6.28 × r
r = 6783 / 6.28
r = 1079.9 miles
Therefore, Diameter of moon is miles is,
d = 2 x r
d = 2 x 1079.9
d = 2159.8 miles
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Five cars start out on a cross-country race. The probability that a car breaks down and drops out of the race is 0.2. Cars break down independently of each other.
(a) What is the probability that exactly two cars finish the race?
(b) What is the probability that at most two cars finish the race?
(c) What is the probability that at least three cars finish the race?
(a) The probability that exactly two cars finish the race is 0.0512.
(b) The probability that at most two cars finish the race is 0.05792.
(c) The probability that at least three cars finish the race is 0.94208.
(a) To determine the probability that exactly two cars finish the race, we have to use binomial distribution. In this case, we have n = 5 trials, and p = 0.8 is the probability that a car finishes the race (1 - 0.2). Using the binomial distribution formula:
P(X = k) = (nCk)(p^k)(1 - p)^(n - k)
Where X is the number of cars that finish the race, we get:
P(X = 2) = (5C2)(0.8²)(0.2)³= (10)(0.64)(0.008)= 0.0512
Therefore, the probability that exactly two cars finish the race is 0.0512.
(b) To determine the probability that at most two cars finish the race, we have to calculate the probabilities of 0, 1, and 2 cars finishing the race and add them up.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)= (5C0)(0.8⁰)(0.2)⁵ + (5C1)(0.8¹)(0.2)⁴ + (5C2)(0.8²)(0.2)³= 0.00032 + 0.0064 + 0.0512= 0.05792
Therefore, the probability that at most two cars finish the race is 0.05792.
(c) To determine the probability that at least three cars finish the race, we can calculate the probability of 0, 1, and 2 cars finishing the race and subtract it from 1, which gives us the probability of at least three cars finishing the race.
P(X ≥ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]= 1 - (0.00032 + 0.0064 + 0.0512)= 0.94208
Therefore, the probability that at least three cars finish the race is 0.94208.
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