The multiplication theorem of probability is if and be two events associated with the sample space . Then, (∩) = ()⋅() and the addition theorem of probability is For two mutually exclusive events ,
(or ) = () + ().
According to the multiplication theorem of probability, the probability of both events and occurring is equal to the product of the probability of occurring and the conditional probability of event occurrence if event happens.
Let and are two events associated with the sample space . ∩ gives the simultaneous occurrence of two events and
We know that the conditional probability of the event given that has occurred is given by (∣) and is given by
(∣) = (∩)(), () ≠ 0
We can write the above equation as
(∩) = (∣)⋅()⋯()
Also, we can write
(∣) = (∩)(), () ≠ 0
(∩) = (∣)⋅()
Combining () and (), we can write
(∩) = (∣)⋅() = (∣)⋅(), where (),() ≠ 0
According to the addition theorem, if two events and , are mutually exclusive, the probability of either or occurring is the total of the individual probabilities of and .
Let event happen in ways, and event happen in ways. Then the total number of happening of events is , which is = +
Thus, the probability of events happening is given by (or ).
By the definition of probability,
() = / and () = /
Adding,
() + () = / + / = + / = / = 1
Therefore,
(or ) = () + ()
--The given question is incomplete; the complete question is
"What is the multiplication and addition theorem of probability? Explain with suitable examples."--
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HELP MEEEEEE PLEASEEEEE ASAPPP
Answer:
Look at my image:
Left 4, down 2
How do you find a 60 degree triangle?.
60° triangle can be calculated in radians as π/3 radians.
60 degree angle is an acute angle in geometry and is expressed as π/3 in radians. Each angle in an equilateral triangle is equal to 60 degrees.
A 60-degree angle is an acute angle since angles less than 90 degrees are considered acute. In the case of a geometric angle, the arc is confined by the sides and is centered at the vertex. When rotated, the arc is centered in the center of rotation and is limited to any other point and its image.
Thus, 60° triangle can be calculated in radians as π/3 radians.
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Use trigonometric identities to simplify each expression
c) sec^2(pi/2-x) (sin^2(x)-sin^4(x))
Using trigonometric identities to simplify the expression, we have sec²(π/2 - x) (sin²(x)-sin⁴(x)) = cos²x
What are trigonometric identities?Trigonometric identities are equations that contain the trigonometric ratios.
How to use trigonometric identities to simplify the expression?Since we have the trigonometric expression sec²(π/2 - x) (sin²(x)-sin⁴(x))
We desire to simplify it as follows.
sec²(π/2 - x) (sin²(x)-sin⁴(x))
Using the trigonometric identity sec²(π/2 - x) = 1/cos²(π/2 - x), we have that
sec²(π/2 - x) (sin²(x)-sin⁴(x)) = 1/cos²(π/2 - x) × (sin²(x)-sin⁴(x))
Also, using the fact that cos²(π/2 - x) = sinx, we have that
1/cos²(π/2 - x) × (sin²(x)-sin⁴(x)) = (sin²(x) - sin⁴(x))/sin²x
Factorizing out sin²x, we have that
(sin²(x) - sin⁴(x))/sin²x = (sin²(x)(1 - sin²(x))/sin²x
= 1 - sin²x
= cos²x
So, sec²(π/2 - x) (sin²(x)-sin⁴(x)) = cos²x
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What is a solution in a system?.
In a system of equations, a solution is a set of values for the variables that make all of the equations in the system true.
For example, consider the system of equations:
3x + 2y = 6
4x - y = 5
A solution to this system would be a set of values for x and y that, when substituted into both equations, make the left side equal the right side. So, for example, one solution could be x = 1, y = 1. If we substitute these values into the equations, we get:
3(1) + 2(1) = 6 and 4(1) - (1) = 5, which is true.
A system of equations can have one unique solution, infinitely many solutions, or no solution.
A system of equations has one unique solution when there is one specific set of values that makes all equations true.
A system of equations has infinitely many solutions when there are infinitely many sets of values that make all equations true. This means that there is a family of solutions, represented by a line or a plane in the case of systems of 2 or 3 equations with 2 or 3 variables.
A system of equations has no solution when there are no values that can be assigned to the variables that make all equations true. This means that the equations are inconsistent.
It's worth noting that a system of equations can be represented in different ways. The solution is the same but the way of finding the solution can change depending on the representation of the system.
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Ai Mi has a loyalty card good for a 13% discount at her local hardware store. What would her total in dollars and cents be, after the discount and before tax, if the total cost of all the items she wants to buy is $39? Round to the nearest cent.
In linear equation, $33.93 would her total in dollars and cents be, after the discount and before tax, if the total cost of all the items she wants to buy is $39.
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to be 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.
Item she wants to buy is priced at $39
After the discount, and before tax, the price is $33.93.
39.00 x .13 = 5.07
39.00 - 5.07 = $33.93
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Use the distributive property to write an expression that is equivalent to 8(a + 4)
Using the distributive property we will find the equivalent expression:
8*(a + 4) = 8a + 32
How to use the distributive property?Remember that the distributive property for the multiplication says that:
A*(B + C) = A*B + A*C
So we distributed the product along the addition.
Here we want to rewrite the expression:
8*(a + 4)
Using the distrubitive property, we can rewrite this as:
8*(a + 4) = 8*a + 8*4
= 8a + 32
That is the equivalent expression.
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For babysitting, Nicole charges a flat fee of $3, plus $5 per hour. Write an equation in slope-intercept form that models this situation.
An equation in slope-intercept form that models this situation :
y = 5x + 3
What is equation ?An equation is a formula in mathematics that expresses equality by connecting two expressions with the equal sign =.
In algebra, an equation is defined as a formula that states that two formulas are equal.
For instance, 3x + 5 = 14 is an equation in which the terms 3x + 5 and 14 are separated by an equal sign.
A mathematical expression used to express the relationship between two terms on opposite sides of a letter.
Like a symbol, there is usually only one variable.
For instance, 2x4 = 2.
A mathematical statement consisting of two expressions joined by an equal sign is known as an equation.
3x - 5 = 16 is an example of an equation.
Solving this equation yields x=7 as the value of the variable x.
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Contruct an iocele triangle in which length of it' equal ide i 6 cm and the angle between them i 75º
A triangle having two equal sides and angles is known as an isosceles triangle. Its equal sides are 6 cm long and make a 75o angle with one another.
A triangle having two equal sides and angles is known as an isosceles triangle. The procedures listed below can be used to build an isosceles triangle with a length of 6 cm and an angle of 75o between the two equal sides:
1. Mark the midway of each of the two 6 cm-long equal lines.
2. Draw a line between the two mid points to create the triangle's base.
3. Beginning at one of the base line's ends, use a compass to draw a 75-degree arc.
4. Create a new arc of Starting from the other end of the base line, measure 75o with a compass.
5. The third corner of the triangle will be formed by the intersection of these arcs.
6. To finish the triangle, connect its three corners with straight lines.
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What is it called when you substitute a number for each variable and simplify?.
When you substitute a number for each variable and simplify it is called a Substitution method.
I will demonstrate how the substitution method works. Suppose we have the following systems of linear equations with two variables:
Equation 1: 3x + 2y = 11
Equation 2: -x + y = 3
By solving these two equations we get x=1 and y=4.
The substitution method is a quick and easy way to algebraically solve a set of linear equations and determine the variables' solutions. According to what the name implies, this method entails determining the value of the x-variable in terms of the y-variable from the first equation and then substituting or replacing the value of the x-variable in the second equation. We can solve and determine the value of the y-variable in this method. Finally, we may use any of the above equations to obtain x by substituting the value of y. This procedure can also be switched around so that we first solve for x before solving for y.
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Do the points 3 2 2 3 and 2 3 form a triangle?.
Yes, the given points (3 ,2) , ( -2 ,- 3 ), and ( 2, 3 ) forms a triangle.
As given in the question,
Given points to form a triangle is given be :
(3 ,2) , ( -2 ,- 3 ), and ( 2, 3 )
Let the triangle name be ABC.
Coordinates of vertex A(x₁ ,y₁ ) = ( 3, 2 )
Coordinates of vertex B(x₂ ,y₂ ) = ( -2, -3 )
Coordinates of vertex C(x₃, y₃) = ( 2, 3 )
General distance formula is :
= √ ( x₂ - x₁ )² + ( y₂ - y₁ )²
Distance between :
AB = √( -3-2)² + (-2 -3)²
= 5√2
BC = √(3+ 3)² + (2 + 2)²
= 2√13
CA = √( 3-2)² + (2 -3)²
= √2
Using Pythagoras theorem we have:
( Hypotenuse )² = ( Side1 )² + ( Side 2)²
Right hand side
= (5√2)² + (√2 )²
= 50 + 2
= 52
= ( 2√13)²
= Left hand side
This implies the triangle formed by the points is right angled triangle.
Therefore, yes the given points forms a right angled triangle.
The above question is incomplete , the complete question is:
Do the points (3 ,2) , ( -2 , -3 ), and ( 2, 3 ) form a triangle?.
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Arthur sits in the park reading a newspaper, then starts walking home at a constant pace. He stops for bagels, then walks the rest of the way home.
Use the graph to answer questions about this scenario.
1. What does the x-axis represent in this situation?
2. What does the y-axis represent?
3. During which parts of the graph does Arthur's distance from home decrease?
4. When is Arthur waiting for bagels? Explain how you know.
5. If Arthur's pace had not been constant after leaving the bagel shop, which part of the graph would change? Explain how that part of the graph would change.
6. If Arthur had spent less time reading his newspaper, which part of the graph would change? How would that part change?
The answers to each part is below..
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is a graph as shown in the image.
The {x} - axis represents the time.The {y} - axis represents the distance from the home.During the 2nd and 4th part.During the 2nd part, he is waiting for bagels.The 2nd part of the graph would change.The first part would have changed.Therefore, the answers to each part is given above.
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HELP MEEEEEEE PLEASEEEEEEEE
Answer:
A to B: Scale of 3, and (10,6)
B to A: Scale of (1/3) and (10,6)
Step-by-step explanation:
See attached worksheet.
graph each function determine which of the functions are linear. find the y-intercept of each function and the slope
The graph of f(x) = 2x + 3 is a linear function and its y-intercept is y = 2x + 3.
What is linear function?Any function is referred to as linear if it represents a straight line on the coordinate plane. For instance, the linear function represented by the equation y = 3x – 2 looks like a straight line on a coordinate plane. Due to the fact that f(x) can substitute for y, this function can be written as f(x) = 3x - 2.
First find the slope of the line
y - y₁ = m(x - x₁)
0 - 3 = m(-1.5 - 0)
m = 2
From the graph, we can take any of the two points i.e.(0, 3) and put it in slope formula
y - y₁ = m(x - x₁)
So,
y - 3 = 2(x - 0)
y = 2x + 3 (y-intercept)
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Full question:
Graph each function determine which of the functions are linear. find the y-intercept of each function and the slope.
f(x) = 2x + 3
what is the exact value of cos 5pi/12
The value of cos 5pi/12 is found to be 0.2588.
Define the term trigonometric ratios?Trigonometric ratios are derived from the value of a ratio of sides of a right-angled triangle and contain the values of all trigonometric functions. These trigonometric ratios of a given acute angle are the ratios of the sides of either a right-angled triangle having respect to that angle.The given trigonometric ratio is-
= cos (5π/12)
The value of π in degrees is 180, substitute it.
= cos (5*180/12)
= cos (900/12)
= cos (75)
Using calculator
≈ 0.2588
Thus, the value of cos 5pi/12 is found to be 0.2588.
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What is product rule example?.
Product rule example is f(x) = (3x - 2x2)(5 + 4x).
Following the Product rule we have
f ' (x) = (3x - 2x2)(0 + 4) + (5 + 4x) (3 - 4x)
= 12x - 8x2 + 15 + 12x - 20x - 16x2
= - 24x2 +4x + 15
Simply put, a product is the outcome of multiplying two functions. This rule, which was developed by Gottfried Leibniz, enables us to compute derivatives that we don't want (or are unable to) multiply quickly. In other words, by combining our understanding of the power rule with the sum and difference rule for derivatives, we can use the product rule to obtain the derivative of two differentiable functions that are being multiplied together.
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Help i want answerrrrrrrr pls
The length of TW is given as follows:
TW = 17.4 units.
How to obtain the length of TW?The length TW is the hypotenuse of a right triangle of sides WU and UT.
The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.
The side lengths are given as follows:
TU = 12.6, as it is tangent just like TV.WU = 12.Hence the length TW is obtained as follows:
(TW)² = 12.6² + 12²
[tex]TW = \sqrt{12.6^2 + 12^2}[/tex]
TW = 17.4 units.
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Can anyone answer this math question for me?
The correct answer is option A (4, 3), 35 units squared.
How to find the vertices?The coordinates of the three vertices of a rectangle are :
A(-3,3) , b(-3,-2) and c(4,-2)
The easy way to find the fourth vertex by the graphing the given points
See the attached figure
the given points are in blue color
and by using the symmetry of the rectangle
So, the coordinates of the fourth point D is (4,3) which are in red color
The base of the rectangle is the distance BC
From the graph ⇒⇒ BC = 7
The height of the rectangle is the distance AB
From the graph ⇒⇒ AB = 5
The area of the rectangle = base * height = 7 * 5 = 35
So, the correct answer is option A(4, 3), 35 units squared.
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Simplify this expression 13+ (-12) - (-5) = ?
Answer: 6
Step-by-step explanation:
When solving this problem, it is important to remember the rules for when we add a negative and subtract a negative. When we add a negative (+ -), we are essentially going to keep the sign attached to the number. In this case, we would still subtract. When subtracting a negative (- -), the signs cancel out, so will add the number instead. With that in mind, we can rewrite the equation and solve it as follows:
13 - 12 + 5
1 + 5
6
Hope this helps!
you go to a store to buy clothes. shirt cost 10.25$ each and pants cost $17 each. let x equal the number of shirts and y equal the number of pants you buy. you can only spend up tp $50 at the store.
Answer:
10.25x+17y ≥ 50
Step-by-step explanation:
10.25x is how much shirts you can buy
17y is how much pants you can buy
50 is the total money you can spend
HOPE THIS HELPED!
The inequality that represents the situation is 10.25x + 17y ≤ 50.
What is an inequality?In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to each other. To solve the inequality, you may multiply or divide each side by the same positive number, add the same amount to each side, take the same amount away from each side, and more. You must flip the inequality sign if you multiply or divide either side by a negative number.
Given:
You go to a store to buy clothes.
The shirt cost 10.25$ each and the pants cost $17 each.
Let x equal the number of shirts and y equal the number of pants you buy. you can only spend up to $50 at the store.
That means,
10.25x + 17y ≤ 50
Therefore, the inequality is 10.25x + 17y ≤ 50.
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Why is the square root of x undefined?.
The square root of negative x undefined is undefined for the set of Real numbers.
We know that the square root of a number is nothing but the factor/value that we can multiply by itself to get that number.
Finding a square root of number is an inverse method of squaring a number.
A radical symbol '√' used to represent the square root of number.
If m is the square root of n, then it is represented as m = [tex]\sqrt{n}[/tex],
We can write above relation as, m² = n
We know that square of any real number is positive.
For example, 3² = (3 × 3) = 9
(-3)² = (-3) × (-3) = 9
i.e., the square root of 9 is 3 and -3
Thus the square root of number is not defined for negative real number.
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The complete question is:
Why is the square root of negative x undefined?.
Please help me with this
Answer:
1 d.p to c 7m
Step-by-step explanation:
you should use phythagoras theorem to caculate the leght of BF ang that is my answer
Tom i 5 year older than jerry. In 2 year, jerry will be three fourth a old a tom. Find their age now
Their age now are 13 years old for Jerry and 18 years old for Tom.
How to find their age?As per data given, Tom age now = 5 + Jerry age now. After time = 2 year, Tom age : Jerry age = 4 : 3. We can determine their age now by substitute methods as follow:
After 2 years = Tom age : Jerry age = 4 : 3
Tom age now + 2 : Jerry age + 2 = 4 : 3
Substitute Tom age now to jerry age now from first formula, so:
Tom age now + 2 : Jerry age now + 2 = 4 : 3
5 + Jerry age now + 2 : Jerry age now + 2 = 4 : 3
3 ( 5 + Jerry age now + 2) = 4 (Jerry age now + 2)
21 + 3 Jerry age now = 4 Jerry age now + 8
Jerry age now = 21 - 8
Jerry age now = 13 years old
Find the tom age from first formula
Tom age now = 5 + Jerry age now
Tom age now = 5 + 13
Tom age now = 18 years old
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Write the explicit function defines the sequence -8, -6, -4, ...?
Answer:
I think it's -2
Step-by-step explanation:
because it keeps going down by 2 and 4-2=2 so its -2
What are the 4 parts of a coordinate plane?.
The four parts of a quadrant are upper right, upper left, lower right, lower left.
The upper right quadrant of the coordinate plane, sometimes known as quadrant 1, has only positive coordinates (QI). Quadrant two is the upper left-fourth of the coordinate plane (QII). Quadrant three is at the bottom left (QIII). Located in the bottom right corner is quadrant number four (QIV). In quadrant I, the x- and y-coordinates are both positive. In quadrant II, the x-coordinate is negative while the y-coordinate is positive. In quadrant III, the x- and y-coordinates are positive. In quadrant IV, the x-coordinate is positive while the y-coordinate is negative. The area enclosed by the x and y axes is referred to as a quadrant in a coordinate system. The quadrants are formed by the 90-degree intersection of the x- and y-axes.
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Kurt is starting a business selling handmade necklaces. He has decided to invest an initial amount of $20 for advertising, and materials cost $7 for each necklace he makes. Kurt can sell his creations for $9 per necklace. Once he makes and sells a certain number of necklaces, he will break even, with identical expenses and sales. How many necklaces would that take? What would the total expenses and sales be then?
PLEASE ANSWER FAST WILL MARK BRAINLIEST!!!
The number of necklaces at the break-even point will be 10.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
Kurt is beginning a business selling high-quality neckbands. He has chosen to contribute an underlying measure of $20 for publicizing, and materials cost $7 for every accessory he makes. Kurt can sell his manifestations for $9 per piece of jewelry.
Let 'x' be the number of necklaces. Then the equations are given as,
Cost = 7x + 20
Sell = 9x
The number of necklaces at the break-even point is calculated as,
Sell = Cost
9x = 7x + 20
2x = 20
x = 10
The number of necklaces at the break-even point will be 10.
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A company sells shirts for $10 each. Its revenue function is r(x) = 10x.
Its costs are $3 per shirt plus $9,000 of fixed costs. Its cost function is
c(x) = 3x + 9000. Use this information for Problems 13-15. The first
one is done for you.
13. Find the company's revenue and costs if it sells 2000 shirts.
Revenue = $20,000; costs = $15,000
14. Profit equals the difference between revenue and cost. Write the
function, p(x), for this company.
15. In 2012, the company sold 5000 shirts. Find its profit.
1
Original content Copyright by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsi
135
a) To find the company's revenue and costs if it sells 2000 shirts, we can substitute 2000 for x in the given revenue and cost functions:
What does a math equation mean?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
Revenue: r(x) = 10x, so r(2000) = 10(2000) = $20,000
Costs: c(x) = 3x + 9000, so c(2000) = 3(2000) + 9000 = $15,000
b) The profit function, p(x), is the difference between the revenue function and the cost function. So, the profit function for this company is:
p(x) = r(x) - c(x) = 10x - (3x + 9000) = 7x - 9000
c) To find the company's profit in 2012, when it sold 5000 shirts, we substitute 5000 for x in the profit function:
p(5000) = 7(5000) - 9000 = $20,000
So, the company had a profit of $20,000 in 2012 when it sold 5000 shirts.
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simplify:cos^2(90°+ß) sin(90°-ß)/cos(360°+ß) sin(360°-ß) sin(180°-ß)
By trigonometric formulas and algebra properties, the trigonometric equation [cos² (90° + β) · sin(90° - β)] / [cos (360° + β) · sin (360° - β) · cos (180° - β)] is equivalent to - 2.
How to simplify a trigonometric equation
In this question we find a trigonometric equation, that is, an equation involving trigonometric functions, that must be simplified by trigonometric formulas and algebra properties. First, write the entire trigonometric formulas:
[cos² (90° + β) · sin(90° - β)] / [cos (360° + β) · sin (360° - β) · cos (180° - β)]
Second, eliminate the cosine squared of the equation by trigonometric formula:
[[1 + cos (180° + 2 · β)] · sin(90° - β)] / [cos (360° + β) · sin (360° - β) · sin (180° - β)]
[sin (90° - β) + cos (180° + 2 · β) · sin(90° - β)] / [cos (360° + β) · sin (360° - β) · sin (180° - β)]
Third, simplify the trigonometric functions for angle additions and subtractions:
[sin (90° - β) + cos (180° + 2 · β) · sin(90° - β)] / [cos (360° + β) · sin (360° - β) · cos (180° - β)]
[cos β - cos 2β · cos β] / [cos β · sin (- β) · sin β]
- (cos β - cos 2β · cos β) / (cos β · sin β · sin β)
Fourth, simplify the expression by algebra properties:
- (1 - cos 2β) / sin² β
(cos 2β - 1) / sin² β
Fifth, eliminate the double angle by trigonometric formulas:
(cos² β - sin² β - sin² β - cos² β) / sin² β
Sixth, simplify the expression by algebra properties:
- 2 · sin² β / sin ² β
- 2 · 1
- 2
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what number has greatest common factor of 2 and is greater than 18 less than 30
Answer:
26
Step-by-step explanation:
The number that has a greatest common factor of 2 and is greater than 18 and less than 30 is 26.
The organizer of a potluck dinner sends 6 people a list of 10 different recipes and asks each person to bring one of the items on the list. If all 6 people randomly choose a recipe from the list, what is the probability that at least 2 will bring the same thing?
The probability of at least two people bringing the same item is 0.19.
What is probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
Given that, The organizer of a potluck dinner sends 6 people a list of 10 different recipes and asks each person to bring one of the items on the list.
The probability of two people bringing the same item can be calculated using the formula P(A∩B) = P(A) × P(B).
In this case, P(A) and P(B) are both 1/10, because each person has the same chance of choosing any one of the 10 items on the list.
Therefore, the probability of two people bringing the same item is 1/100. To calculate the probability of at least two people bringing the same item, we can use the formula P(A∪B) = P(A) + P(B) - P(A∩B).
In this case, P(A∪B) = 1/10 + 1/10 - 1/100 = 0.19.
Hence, the probability of at least two people bringing the same item is 0.19.
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What does the function f 14 )= 11?.
Function f(14) = 11 shows that the outcome of a function at x = 14 the value of the function is 11.
Function is a mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences. The German mathematician Peter Dirichlet initially offered the contemporary definition of function in 1837:
To specify what the function f (14) = 11 means.
Here,
f( 14 ) = 11
f( 14 ) = f( x )
x = 14
f( x ) = 11
Implies that,
f(x) = y is function, when x = 14 the function gives y = 11 ,,means that for x = 14 the outcome of the function is 14.
Therefore, function f(14) = 11 implies that the outcome of a function at x = 14 the value of the function is 11.
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