[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=12\\ V=1281.12 \end{cases} \implies 1281.12=\cfrac{\pi (12)^2 h}{3} \\\\\\ (3)(1281.12)=144\pi h\implies \cfrac{(3)(1281.12)}{144\pi }=h\implies 8.50\approx h[/tex]
If the parent cubic function, f(x) = x3, is transformed to F(x) = 1/2x^3+2 what will be the effect on the graph of the parent function?
A.The graph will shift 2 units left and be vertically compressed so the graph will appear wider.
B.The graph will shift 2 units right and will be vertically compressed so that the graph will appear narrower.
C.The graph will shift 2 units up and will be vertically compressed so that the graph will appear wider.
D.The graph will shift 2 units up and will be vertically stretched so that the graph will appear narrower.
The effect on the graph of the parent function after transformation is:
C.The graph will shift 2 units up and will be vertically compressed so that the graph will appear wider.
What is the effect on the graph of the parent function?The vertical shift depends on the value of k. The vertical shift is described as:
f(x) = f(x) + k - The graph is shifted up k units.
f(x) = f(x) − k - The graph is shifted down k units.
Now, we are told that f(x) = x³, is transformed to f(x) = ¹/₂x³ + 2
Thus, the graph is shifted up by 2 units.
Compressing and stretching depends on the value of a.
When a is greater than 1: Vertically stretched
When a is between 0 and 1: Vertically compressed
In this case, a is 1/2 which is between 0 and 1 and so is Vertically compressed
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Which describes a possible independent variable for the given dependent variable?
Your monthly cell phone bill
1. Who you call most often each month
2. The amount of data you use each month
3. The color of your phone
4. The amount of money you spend for lunch each month
Answer:
1
explanation in head
The distribution of all registered nurses' salaries on the Treasure Coast is known to be normally distributed
with a mean of $50, 650 and a standard deviation of $1,000. Use this information to determine the
following two probabilities. Round solutions to four decimal places, if necessary.
The probability that a single randomly selected nurse's salary is greater than $50,516 is 0.5533 and the probability that a random sample of 95 nurses have a salary greater than $50,516 is 0.9098.
What is the probability that a single randomly selected nurse's salary is greater than $50,516a. To find the probability that a single randomly selected nurse's salary is greater than $50,516, we need to standardize the value using the mean and standard deviation of the distribution, and then use a standard normal table or calculator to find the probability.
The standardized value (z-score) is:
z = (x - μ) / σ = (50,516 - 50,650) / 1,000 = -0.134
Using a standard normal table or calculator, we can find the probability that a randomly selected nurse's salary is greater than $50,516:
P(x > 50,516) = P(z > -0.134) = 0.5517
Therefore, the probability that a single randomly selected nurse's salary is greater than $50,516 is 0.5533.
b. To find the probability that a random sample of 95 nurses have a salary greater than $50,516, we need to use the central limit theorem, which states that the distribution of the sample means approaches a normal distribution with mean μ and standard deviation σ/√n, where n is the sample size.
The mean of the sample means is still μ = 50,650, but the standard deviation of the sample means is now:
σ/√n = 1,000 / √95 = 102.06
We want to find the probability that the sample mean is greater than $50,516:
P(x > 50,516) = P(z > (50,516 - 50,650) / (1,000 / √95)) = P(z > -1.335)
Using a standard normal table or calculator, we can find the probability:
P(x > 50,516) = P(z > -1.335) = 0.9098
Therefore, the probability that a random sample of 95 nurses have a salary greater than $50,516 is 0.9098.
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1. The table to the left shows the joint probability function for X and Y . a. Explain why this is a legitimate joint probability function for X and Y . X b. Find p(1,2) . c. Find P(X<1,Y≥2) . d Find p x (3)
The joint probability function for X and Y is legitimate because the sum of all probabilities is equal to 1 and the probabilities are non-negative. Each point in the domain has a corresponding probability.b. p(1,2) means the probability of X=1 and Y=2. p_x(3) = 0.1 + 0.05 = 0.15.
a. Probability function is considered legitimate when the sum of all the probabilities is 1, the probabilities are non-negative and each point in the domain must have a corresponding probability. Here, the joint probability function for X and Y is legitimate because the sum of all probabilities is equal to 1 and the probabilities are non-negative. Each point in the domain has a corresponding probability.b. p(1,2) means the probability of X=1 and Y=2. We can see from the table that p(1,2) = 0.04.c. P(X<1, Y≥2) means the probability of X being less than 1 and Y being greater than or equal to 2. We can find the probabilities by adding up all the probabilities in the cells that meet this condition. From the table, we can see that the cells (0,2) and (0,3) meet this condition. Therefore, P(X<1, Y≥2) = 0.01 + 0.01 = 0.02.d. We need to find p_x(3), which means the probability of X=3. We can find this by adding up all the probabilities where X=3. From the table, we can see that the cells (3,1) and (3,2) meet this condition. Therefore, p_x(3) = 0.1 + 0.05 = 0.15.
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Suppose that 10% of all homeowners in an earthquake-prone area of California are insured against earthquake damage. Four homeowners are selected at random; let x denote the number among the four who have earthquake insurance.
(a) Find the probability distribution of x. (Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.1)(.9)(.1)(.1) and associated x value of 3. There are 15 other outcomes.)
Value of x Probability
0 1 2 3 4 (b) What is the most likely value of x?
(a) 0
(b) 1
(c) 0 and 1
(d) 3
(e) 4
(C) What is the probability that at least two of the four selected homeowners have earthquake insurance?
P (at least 2 of the 4 have earthquake insurance) =
Answer: (a) The possible outcomes and their probabilities are:
Value of x Probability
0 0.6561 (0 F's and 4 S's)
1 0.2916 (1 F and 3 S's, or 2 F's and 2 S's, or 3 F's and 1 S)
2 0.0486 (2 F's and 2 S's)
3 0.0036 (1 S and 3 F's)
4 0.0001 (4 F's and 0 S's)
(b) The most likely value of x is the one with the highest probability, which is x = 0.
(c) The probability that at least two of the four selected homeowners have earthquake insurance is equal to 1 minus the probability that 0 or 1 of them have earthquake insurance:
P(at least 2 of the 4 have earthquake insurance) = 1 - P(x = 0) - P(x = 1)
= 1 - 0.6561 - 0.2916
= 0.0523
Therefore, the probability that at least two of the four selected homeowners have earthquake insurance is 0.0523.
Enjoy!!!!!
Probability distribution of x is calculated by finding probabilities of each possible outcome, the most likely value of x is 0 with a probability of 0.6561, probability of at least two homeowners having earthquake insurance is 0.0523.
(a) The probability distribution of x can be found by calculating the probabilities of each possible outcome (0, 1, 2, 3, 4 homeowners with insurance) and associating them with their respective x values.
Value of x | Probability
--- | ---
0 | (.9)(.9)(.9)(.9) = 0.6561
1 | (.1)(.9)(.9)(.9) + (.9)(.1)(.9)(.9) + (.9)(.9)(.1)(.9) + (.9)(.9)(.9)(.1) = 0.2916
2 | (.1)(.1)(.9)(.9) + (.1)(.9)(.1)(.9) + (.1)(.9)(.9)(.1) + (.9)(.1)(.1)(.9) + (.9)(.1)(.9)(.1) + (.9)(.9)(.1)(.1) = 0.0486
3 | (.1)(.1)(.1)(.9) + (.1)(.1)(.9)(.1) + (.1)(.9)(.1)(.1) + (.9)(.1)(.1)(.1) = 0.0036
4 | (.1)(.1)(.1)(.1) = 0.0001
(b) The most likely value of x is 0, as it has the highest probability (0.6561).
(c) The probability that at least two of the four selected homeowners have earthquake insurance can be found by adding the probabilities of the outcomes with 2, 3, and 4 homeowners with insurance:
P (at least 2 of the 4 have earthquake insurance) = 0.0486 + 0.0036 + 0.0001 = 0.0523
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a little stuck on this, please help!!
A prism has a height of 6.1 meters. If the side of the base is 4.6 m and the apothem is 3.17 meters. Find the volume of the prism
The volume of the prism is 222. 4 m³
How to determine the volume of the prismThe formula for calculating the volume of a pentagonal prism is expressed as;
V = [1/2 x 5 x b x a] x h of the prism
Given that the parameters are;
V is the volume of the prismb is the base side of the prisma is the apothem of the prismh is the height of the prismNow, substitute the values, we get;
Volume , = ( 1/2 × 5 × 4.6 × 3.17) × 6.1
Multiply the values
Volume , V = (72.91/2) × 6.1
divide the values in the bracket
Volume = 36. 46 × 6.1
Multiply the values
Volume = 222. 4 m³
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Which option is the answer?
The correct statement regarding the association between the variables is given as follows:
C. A student who feels some pressure from homework is most likely to prefer to be rich.
How to interpret the association between the variables?The two variables for this problem are defined as follows:
Preferred status.Pressure from homework.For the people who feel some pressure from homework, the preferred status are given as follows:
Happy: 0.39.Healthy: 0.31.Rich: 0.50.As rich has the highest proportion, the student is most likely to prefer to be rich, hence option C is the correct option.
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. A person has a beginning balance of $660. She pays $90 on the 9th day, and she charges
320 on the 28th day. What amount of interest is due on her account if it has an APR of 26
ercent?
a.
b.
$7.48
$19.02
C. $20.79
d. $6.71
e. $13.38
To calculate the interest due, we first need to determine the average daily balance for the month.
From the beginning of the month until the 9th day, the balance is $660. From the 10th to the 27th, the balance is $660 - $90 = $570. On the 28th, the balance increases to $570 + $320 = $890.
So the average daily balance is:
[(31 - 9) x $570 + 9 x $660 + 22 x $890] / 31 = $691.29
Next, we need to calculate the monthly interest rate:
26% APR = 0.26 / 12 = 0.02167
Finally, we can calculate the interest due:
$691.29 x 0.02167 = $14.98
Therefore, the answer is closest to option (a) $7.48.
The tuition at a private college is increasing from $52,500 to $60,500. Find the absolute change and
relative change in tuition.
Absolute change:
Relative change:
Round to the nearest tenth of a percent and don't forget to include a percent sign, %, in your answer.
Answer: the absolute change in tuition is $8,000 and the relative change in tuition is 15.2%.
Step-by-step explanation: The absolute change in tuition is the difference between the new tuition and the old tuition. Therefore:
Absolute change = New tuition - Old tuition
Absolute change = $60,500 - $52,500
Absolute change = $8,000
The relative change in tuition is the ratio of the absolute change to the old tuition, expressed as a percentage. Therefore:
Relative change = (Absolute change / Old tuition) x 100%
Relative change = ($8,000 / $52,500) x 100%
Relative change = 15.2%
Many new cars provide detailed information about engine performance on the dashboard. One such feature allows drivers to observe current fuel efficiency, recorded in miles per gallon, as they drive. A consumer takes a long trip driving at different speeds, while a passenger records both driving speed in miles per hour and fuel efficiency for a number of selected points along the trip. A least-squares equation that relates speed to fuel efficiency is given by .
Based on the residual plot shown, is a linear model appropriate for comparing driving speed and fuel efficiency?
A linear model is appropriate because the residual plot is clearly curved.
A linear model is not appropriate because the residual plot shows a clear pattern.
A linear model is appropriate because the residuals are decreasing at higher car speeds.
A linear model is not appropriate because there are more negative residuals than positive residuals.
A linear model is not appropriate because the residual plot shows a clear pattern.
What is a residual plot, and how may it be used to judge a linear model's reliability?The residuals (i.e., the discrepancies between the actual observed values and the anticipated values) are shown against the independent variable in a residual plot (i.e. the variable that the model is trying to predict). The x-variable is often the independent variable in a linear model.
The residual plot displays a curved pattern, which indicates that comparing driving speed and fuel economy should not be done using a linear model. This suggests that there may be other factors besides speed that have an impact on fuel economy and that the connection between speed and fuel efficiency is not strictly linear.
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PLEASE SOMEBODY HELP ME
Answer:
1. rhombus and square
2. rhombus, rectangle, square
3. rectangle and square
4. rectangle, squares, rhombus
5. same as 3
6. rhombus, rectangle, square
7. rhomus, rectangle, square
8. rectangle, square
Step-by-step explanation:
Dina has a mass of 50 kilograms and is waiting at the top of a ski slope that’s 5 meters high. The maximum kinetic energy she can reach when she skis to the bottom of the slope is joules. Use pe = m × g × h and g = 9. 8 m/s2. Ignore air resistance and friction
The maximum kinetic energy Dina can reach is 250 joules, calculated by multiplying her mass of 50 kg, gravitational acceleration of 9.8 m/s2, and the height of the slope of 5 m.
50 kg x 9.8 m/s2 x 5 m = 250 joules
The maximum kinetic energy that Dina can reach when she skis to the bottom of the slope is calculated by the equation pe = m × g × h. This equation states that the potential energy of an object is equal to its mass multiplied by the gravitational acceleration, which is 9.8 m/s2, and the height of the slope. In Dina’s case, her mass is 50 kg and the height of the slope is 5 m, so the potential energy is equal to 50 kg x 9.8 m/s2 x 5 m, which is equal to 250 joules. This means that the maximum kinetic energy Dina can reach when she skis to the bottom of the slope is 250 joules. This equation is valid as long as air resistance and friction are both ignored, as these can have a significant effect on the kinetic energy of an object.
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Find all the cube numbers greater than 20 but less than 50
the only cube number between 20 and 50 is 27.
Step 1: Find the smallest cube greater than 20.
To find the smallest cube greater than 20, we can start by checking the cube of the smallest integer, which is 1. Since [tex]1^3[/tex]= 1, which is less than 20, we move on to the cube of the next integer, which is 2. We find that [tex]2^3[/tex]= 8, which is still less than 20. Finally, we check the cube of the next integer, which is 3. We find that [tex]3^3[/tex]= 27, which is the smallest cube greater than 20.
Step 2: Find the largest cube less than 50.
To find the largest cube less than 50, we can start by checking the cube of the largest integer that is less than the cube root of 50. The cube root of 50 is approximately 3.68, so the largest integer less than the cube root of 50 is 3. We find that [tex]3^3[/tex] = 27, which is less than 50. Since the next cube, [tex]4^3[/tex] = 64, is greater than 50, we know that [tex]3^3[/tex] is the largest cube less than 50.
Step 3: List all the cubes between the smallest cube greater than 20 and the largest cube less than 50.
Now that we have found the smallest cube greater than 20 and the largest cube less than 50, we can list all the cubes between them. The cubes between 27 and 27 are just 27, so we have:27
Therefore, the only cube number between 20 and 50 is 27.
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E- Education is A new start-up that develops and markets MBA courses offered over the internet
The startup E-Education can go according to option 2 and option 3. Both the options are giving returns of $1500000 for 2 years.
There are three options available with E-Education to fulfill the company's need for additional space.
The decision tree for the available options is made below;
Option1 Option2 Option3
$ $ $
Cost of Moving to New Building 1000000 2000000
Leasing cost 1000000 1300000 1500000
Total Cost of the Option 2000000 1500000 1500000
Conclusion: The startup E-Education can go according to option 2 and option 3. Both the options are giving returns of $1500000 for 2 years.
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Amanda is buying her best friend a CD for
her upcoming birthday. The CD is
originally priced at $22.50, but has a sale
sticker for 10% off. How much will
Amanda save on her friend's gift?
Answer:
$2.25
Step-by-step explanation:
Answer:
Step-by-step explanation:
02.25 is your answer because 10% off of 22.50 is 20.25:)
Let the demand function for a good be Q = 75 - 3P. Determine the price elasticity of
demand when P=15. Give your answer to one (1) decimal place
The price elasticity of demand when P=15 is -0.9
To find the price elasticity of demand when P=15, we need to calculate the percentage change in quantity demanded and the percentage change in price. Let's start with the percentage change in quantity demanded.
ΔQ/Q = (Q2 - Q1)/Q1
where Q1 is the initial quantity demanded and Q2 is the new quantity demanded.
Substituting the values from the demand function, we get:
ΔQ/Q = [(75 - 3P2) - (75 - 3P1)]/(75 - 3P1)
where P1 = 15 and P2 = P1 + ΔP = 15 + 15% = 17.25 (since ΔP/P = 15%).
ΔQ/Q = [(75 - 3(17.25)) - (75 - 3(15))]/(75 - 3(15))
ΔQ/Q = (-8.25)/60
ΔQ/Q = -0.1375
Therefore, the percentage change in quantity demanded is -13.75%.
Now, let's calculate the percentage change in price:
ΔP/P = (P2 - P1)/P1
ΔP/P = (17.25 - 15)/15
ΔP/P = 0.15
Therefore, the percentage change in price is 15%.
Substituting these values in the formula for price elasticity of demand, we get:
E = (-0.1375) / 0.15
E = -0.9167
The price elasticity of demand when P=15 is approximately -0.9.
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DIG DEEPER! You use a crowdfunding website to raise money. The website keeps 5% of each donation. Five of your friends each donate the same amount. The total funding you receive is $47.50. How much does each friend donate?
Please help step by step
Answer:
MARK BRIANLIST!!
Question 11 Find the average rate of change of g(x)=5x^(3)+(9)/(x^(2)) on the interval -1,1. Question Help: Video Submit Question
The average rate of change of g(x) on the interval [-1, 1] is 5.
The average rate of change of a function over a given interval is the difference between the values of the function at the endpoints of the interval, divided by the length of the interval. Here, we have to Find the average rate of change of g(x) = 5x³ + 9/x² on the interval [-1, 1].Now we'll apply the formula: (g(1) - g(-1)) / (1 - (-1)) (g(1) - g(-1)) / 2 We have g(x) = 5x³ + 9/x². Hence g(1) = 5(1)³ + 9/(1)² = 5 + 9 = 14 g(-1) = 5(-1)³ + 9/(-1)² = -5 + 9 = 4 Therefore, the average rate of change of g(x) on the interval [-1, 1] is: (14 - 4) / 2 = 10 / 2 = 5 Therefore, the average rate of change of g(x) on the interval [-1, 1] is 5.
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thr radius of a circle is 1 meter, What is the length of a 45° arc?
Answer:
0.785
Step-by-step explanation:
formula = (a/360)2r π
Find the area of the figure
Answer:
The area of the figure is 90 yd²
Step-by-step explanation:
Dividing the shape into three leading to 3 rectangles
The area of a rectangle is length * breadth;
First section + Second section + Third section
(9 * 3) + ( 9 * 4) + (9 * 3) = 27 + 36 + 27 = 90 yd²
Answer:
74yd²
Step-by-step explanation:
You can divide into 3 rectangles from bottom down
Area = length x width
1st rectangle = 9 x 3 = 27yd²
2nd rectangle = 5x 4 = 20yd²
3rd rectangle = 9 x 3 = 27yd²
Add all the area
27 +20+27 = 74yd²
Another method
Find the area of the whole rectangle with the white area closed and included
Lenght 10yd, width 9yd
Area = 10 x9 = 90yd²
Find the area of the white area only
A = 4 x4 =16yd ²
Subract the area of the white square from the whole area of the rectangle
90 - 16 =74yd²
A table lamp emits light in the shape of a hyperbola. If the hyperbola is modeled by the equation 9x2 – 16y2 + 576 = 0, which of the following equations represents the boundaries of the light?
After answering the given query, we can state that The light's limits are equation therefore represented by the equations y = (3/4)x and y = -(3/4)x.
What is equation?A mathematical statement known as an equation demonstrates the equality of two expressions when they are joined by the equals symbol ('='). As an illustration, 2x - 5 = 13. 2x-5 and 13 are examples of expressions. The letter '=' joins the two phrases together. An equation is a mathematical formula with two algebraic expressions on either side of the equal symbol (=). It shows how the left and right formulas have an equivalent connection. In any formula, L.H.S. equals R.H.S. (left side = right side).
We must rewrite the provided equation in terms of y in order to ascertain the equation denoting the boundaries of the light.
Starting with the expression provided:
[tex]9x^2 - 16y^2 + 576 = 0\\-16y^2 + 576 = -9x^2\\y^2 - 36 = (9/16)x^2[/tex]
When both edges are squared:
y = ±(3/4)x
The light's limits are therefore represented by the equations y = (3/4)x and y = -(3/4)x.
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Solve each inequality.
a. 9t-5(t - 5) ≤ 4(t - 3)
Answer:
No solution
Step-by-step explanation:
9t-5(t - 5) ≤ 4(t - 3)
9t - 5t + 25 ≤ 4t - 12
4t + 25 ≤ 4t - 12
0 ≤ -37
This is not true, so there is no solution to this inequality.
Fill in the blanks below in order to justify
whether or not the mapping shown
represents a function.
The mapping above represents a function since each element in Set A maps to exactly one element in Set B where there is no repetition or ambiguity in the mapping where there are no two distinct elements in Set A that map to the same element in Set B.
Describe Sets?Sets can be used to represent a wide range of mathematical and real-world concepts, such as the set of prime numbers, the set of colors in a rainbow, or the set of people who have visited a particular city.
Sets are usually described using set-builder notation, which uses a rule to define the set. For example, the set of even numbers can be described as {x | x is an integer and x is even}, where the vertical bar | means "such that" and the condition "x is an integer and x is even" specifies the elements of the set.
The mapping above represents a function since each element in Set A maps to exactly one element in Set B, where there is no repetition or ambiguity in the mapping.
To fill in the blanks:
The mapping above represents a function since each element in Set A maps to exactly one element in Set B where there is no repetition or ambiguity in the mapping where there are no two distinct elements in Set A that map to the same element in Set B.
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Work out
25% of 401. 16m
Give your answer rounded to 2 DP
The 25 percent of 401.16m is equal to 100.29m, rounded to 2 decimal places as 100.30m.
To calculate 25% of 401.16m, we can use the formula:
percentage × value = result
where the percentage is 25%, the value is 401.16m, and the result is what we want to find.
So, we can write:
25% × 401.16m = (25/100) × 401.16m = 0.25 × 401.16m = 100.29m
Therefore, 25% of 401.16m is equal to 100.29m.
To round this to 2 decimal places, we need to look at the third decimal place, which is 9. Since 9 is greater than or equal to 5, we round up the second decimal place, which is 0. Therefore, the final answer rounded to 2 decimal places is:
100.29m rounded to 2 decimal places is 100.30m.
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Rates and Ratios 1. A motorist covers a distance of 360 km in .exactly 4 hour Appropriately how far did the motorist drive in one hour
The motorist drove covered a distance of 90 kilometers in one hour.
How far did the motorist drive in one hour?Speed is simply referred to as distance traveled per unit time.
Mathematically, it is expressed as;
Speed = Distance ÷ time.
To determine how far the motorist drove in one hour, we need to divide the total distance covered by the total time taken:
distance in one hour = total distance / total time
Given that;
distance covered = 360 kmtime taken = 4 hoursSubstituting the given values, we get:
distance in one hour = 360 km / 4 hours
distance in one hour = 90 km/hour
Therefore, the motorist drove at a speed of 90 kilometers per hour.
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pelcentile a cumulative frequency curve; the value that would be sampled 95 out of 100 times a frequency polygon; the value in the dataset that is most likely to occur question 9 choose the best answer. which would be a uniform probability distribution? the probability of reaching a temperature of 75f on any given day of the year in st. louis, mo a time period in which it rained 25% of the time and did not rain 75% of the time the probabilities of drawing any individual card in a deck with one draw flipping a coin two times and recording whether heads or tails
As per the given options, the uniform probability distribution can be defined as the probabilities of drawing any individual card in a deck with one draw.
A uniform distribution is a statistical probability function that assigns equal probability across the distribution's entire range. For example, when rolling a fair die, each of the six outcomes has an equal probability of 1/6, which is a uniform probability distribution.
The formula for a Uniform Distribution.The probability density function of the uniform distribution is:f (x) = {1 / (b - a)} for a ≤ x ≤ bWhere, a = lower limitb = upper limitx = random variablef (x) = probability density function.
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[tex]2^{3x} =11[/tex]. Find the value of x.
Answer:
[tex] \frac{ log_{2}(11) }{3} [/tex]
Step-by-step explanation:
[tex]3x = log_{2}(11) [/tex][tex]
x = \frac{ log_{2}(11) }{3} [/tex]
A survey found that 10% of people believe that they have seen a UFO. Choose a sample of 15 people at random. Find the probability of the following. Round intermediate calculations and final answers to at least three decimal places.
(a) At least 3 people believe that they have seen a UFO.
(b) 3 or 4 people believe that they have seen a UFO.
(c) Exactly 4 people believe that they have seen a UFO
(a) Prοbability οf at least 3 peοple believe that they have seen a UFO 0.857.
(b) Prοbability οf 3 οr 4 peοple believe that they have seen a UFO 0.181.
(c) Prοbability οf Exactly 4 peοple believe that they have seen a UFO 0.00049 (rοunded tο 3 decimal places).
What is prοbability?Prοbability is a branch οf mathematics that deals with the study οf randοmness and uncertainty in events. It is the measure οf the likelihοοd οr chance that an event will οccur. Prοbability is expressed as a number between 0 and 1, where 0 indicates that the event will nοt οccur and 1 indicates that the event will οccur with certainty.
This is a binοmial prοbability prοblem, where the prοbability οf success (p) is 0.1 and the sample size (n) is 15.
(a) At least 3 peοple believe that they have seen a UFO:
Using the cοmplement rule, the prοbability οf having less than 3 peοple whο believe they have seen a UFO is:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = (0.9)¹⁵ + 15(0.1)(0.9)¹⁴ + (105)(0.1)²(0.9)¹³
where X is the number οf peοple whο believe they have seen a UFO in the sample. Therefοre, the prοbability οf having at least 3 peοple whο believe they have seen a UFO is:
P(X ≥ 3) = 1 - P(X < 3) = 1 - [(0.9)¹⁵ + 15(0.1)(0.9)¹⁴ + (105)(0.1)²(0.9)¹³] = 0.170
Sο the prοbability οf at least 3 peοple believing they have seen a UFO is 0.170.
(b) 3 οr 4 peοple believe that they have seen a UFO:
Using the binοmial prοbability fοrmula, we can calculate the prοbabilities fοr 3 and 4 peοple believing they have seen a UFO and then add them tοgether:
P(X = 3) = (15 chοοse 3)(0.1)³(0.9)¹² = 0.185
P(X = 4) = (15 chοοse 4)(0.1)⁴f(0.9)¹¹ = 0.099
Therefοre, the prοbability οf having 3 οr 4 peοple whο believe they have seen a UFO is:
P(3 οr 4) = P(X = 3) + P(X = 4) = 0.185 + 0.099 = 0.284
Sο the prοbability οf 3 οr 4 peοple believing they have seen a UFO is 0.284.
(c) Exactly 4 peοple believe that they have seen a UFO:
Using the binοmial prοbability fοrmula, we can calculate the prοbability fοr 4 peοple believing they have seen a UFO:
P(X = 4) = (15 chοοse 4)(0.1)⁴ (0.9)¹¹ = 0.099
Sο the prοbability οf exactly 4 peοple believing they have seen a UFO is 0.099.
Hence,
(a) P(X ≥ 3) = 0.857
(b) P(3 ≤ X ≤ 4) = 0.181
(c) P(X = 4) = 0.00049 (rοunded tο 3 decimal places)
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