Answerhshxbx
Ste-by-step explanation:
Hwsjbb
Suppose the round-trip airfare between Boston and Orlando follows the normal probability distribution with a mean of $387.20 and a standard deviation of $68.50. What is the probability that a randomly selected airfare between Boston and San Francisco will be less than $300
The probability that a randomly selected airfare between Boston and Orlando will be less than $300 is approximately 10.04%.
To find the probability that a randomly selected airfare between Boston and Orlando will be less than $300, we need to calculate the z-score and use the standard normal distribution table.
Calculate the z-score: z = (x - mean) / standard deviation = (300 - 387.20) / 68.50 = -1.2768. Look up the corresponding z-score in the standard normal distribution table. The area to the left of z = -1.28 is 0.10035.Therefore, the probability that a randomly selected airfare between Boston and Orlando will be less than $300 is approximately 0.10035, or 10.04%.
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Write a rule for the nth term of the geometric sequence if r = 1/4 and a3 = 2.
Answer:
[tex]a_{n} = 32(\frac{1}{4})^{n-1}[/tex]
Step-by-step explanation:
The nth term of a geometric sequence is given by the following equation.
[tex]a_{n+1} = ra_{n}[/tex]
In which r is the common ratio.
This can be expanded for the nth term in the following way:
[tex]a_{n} = a_{1}r^{n-1}[/tex]
In which [tex]a_{1}[/tex] is the first term.
This means that for example:
[tex]a_{3} = a_{1}r^{3-1}[/tex]
So
[tex]a_{3} = a_{1}r^{2}[/tex]
[tex]2 = a_{1}(\frac{1}{4})^{2}[/tex]
[tex]2 = \frac{a_{1}}{16}[/tex]
[tex]a_{1} = 32[/tex]
Then
[tex]a_{n} = 32(\frac{1}{4})^{n-1}[/tex]
A small appliance manufacturer is planning to open a repair facility that would receive broken appliances from customers and repair them. Customers would send their broken appliances to the facility, and when repaired, they would be returned directly to the customer. There are five different appliance models that would be repaired, identified here as A, B, C, D, and E. The industrial engineering department has provided estimates of the average time to repair each model. These mean times are, respectively, 23 min, 42 min, 19 min, 27 min, and 35 min. The management wants to staff the facility in anticipation of a weekly return rate of 100 appliances per week for models A, B, and C, and a weekly return rate of 150 appliances for models D and E. The facility will operate one shift five days per week, but it is anticipated that each repairperson will spend an average of only 6.0 hours per day repairing appliances. How many repairpersons will be required for the facility
Answer:
There will be required 590 repair persons.
Step-by-step explanation:
Lets count the amount of minutes that are going to be used per week repairing. The amount of minutes of repair work required per week is
100*23 + 100*42+ 100*19 + 150*27 + 150*35 = 17700.
Each worker works 6 hours per day in average and 5 times per week. This means that each worker will do 30 hours of work each week. Therefore, we will need 17700/30 = 590 workers (In practice you should contract a few more, like 600, just in case).
In a chess tournament, each player plays every other player exactly once. If it is known that 105 games were played, how many players were there in the tournament?
Answer:
15 players played in the chess tournament.
Step-by-step explanation:
When there are 15 players, the first player plays 14 games and step aside.
remaining 14 player
next player plays 13 games. steps aside
remaining 13 players
next player plays 12 games. steps aside
remaining 12 players
next player plays 11 games. steps aside
remaining 11 players
.
.
.
.
Last player plays 1 game and steps aside
remaining 1 player who will not play against himself.
∴ Sum of all games played
= 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 105 games played
Answer:
Step-by-step explanation:
Let be the number of players. There was a game for every pair of players, so there must be 105 pairs of players.
(−1)/2=105.
should be easy to solve after that.
what is the greatest common factor "GCF" of 24x^2 y^3 z^4 + 18x^6 z - 36x^3 y^2
Answer:
6x^2
Step-by-step explanation:
What is Four times Seven?
Answer:
28
Step-by-step explanation:
thats math
Which figure cannot be made by slicing through a cube? Select all that apply.
A
a circle
B
an octagon
C
a rectangle
D
a square
E
a decagon
F
a triangle
Answer:
A. a circle, B. an octagon, E. a decagon, F. a triangle
Step-by-step explanation:
Squares and rectangles can be made by slicing a cube.
Answer:
A
a circle
B
an octagon
E
a decagon
Step-by-step explanation:
I had this question on my math test and these are correct
What is the answer to 2(4+9w)-4
Answer:
=2(4+9w)
=2(4) 2(+9w)
=8 +18w
=18w+8
Step-by-step explanation:
Consider the function Upper F (x comma y )equalse Superscript negative x squared divided by 6 minus y squared divided by 6 and the point Upper P (negative 3 comma 3 ). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P.
Answer:
a.
[tex]\vec{u}=\frac{\bigtriangledown F(-3,3)}{|\bigtriangledown F(-3,3)|}=\frac{1}{\sqrt{2}}[\hat{i}-\hat{j}][/tex] (ascent)
[tex]\vec{u}=-\frac{\bigtriangledown F(-3,3)}{|\bigtriangledown F(-3,3)|}=-\frac{1}{\sqrt{2}}[\hat{i}-\hat{j}][/tex] (descent)
b.
[tex]\vec{v}=\frac{1}{\sqrt{2}}[\hat{i}+\hat{j}][/tex]
Step-by-step explanation:
a. The function is given by:
[tex]F(x,y)=e^{-(x^2/6+y^2/6)}[/tex]
the point is P(-3,3)
a. The unit vector that gives the direction of the steepest ascent is necessary to compute the gradient of F(x,y):
[tex]\bigtriangledown F(x,y)=e^{-(x^2/6+y^2/6)}(-\frac{x}{3})\hat{i}+e^{-(x^2/6+y^2/6)}(-\frac{y}{3})\hat{j}\\\\\bigtriangledown F(x,y)=-\frac{1}{3}e^{-(x^2/6+y^2/6)}[x\hat{i}+y\hat{j}][/tex]
The, it is necessary to evaluate in the point P, and to compute the norm of the vector in order to get the unit vector:
[tex]\bigtriangledown F(-3,3)=-\frac{1}{3}e^{-(\frac{9}{6}+\frac{9}{6})}[-3\hat{i}+3\hat{j}]\\\\\bigtriangledown F(-3,3)=e^{-3}[\hat{i}-\hat{j}]\\\\|\bigtriangledown F(-3,3)|=\sqrt{(e^{-3})^2+(e^{-3})^2}=\sqrt{2}e^{-3}\\\\\vec{u}=\frac{\bigtriangledown F(-3,3)}{|\bigtriangledown F(-3,3)|}=\frac{1}{\sqrt{2}}[\hat{i}-\hat{j}][/tex] (ascent)
for the steepest descend you have
[tex]\vec{u}=-\frac{\bigtriangledown F(-3,3)}{|\bigtriangledown F(-3,3)|}=-\frac{1}{\sqrt{2}}[\hat{i}-\hat{j}][/tex]
b.
the vector with the direction of no change is a vector perpendicular to grad(F):
[tex]\bigtriangledown F(-3,3)\cdot \vec{v}=0\\\\e^{-3}v_1-e^{-3}v_2=0\\\\v_1=v_2[/tex]
furthermore, v is an unit vector:
[tex]\sqrt{v_1^2+v_2^2}=1\\\\v_1 ^2+v_1^2=1\\\\2v_1^2=1\\\\v_1=\frac{1}{\sqrt{2}}=v_2[/tex]
then, the vector is:
[tex]\vec{v}=\frac{1}{\sqrt{2}}[\hat{i}+\hat{j}][/tex]
A 29-m tall building casts a shadow. The distance from the top of the building to the tip of the shadow is 34 m. Find the length of the shadow. If necessary, round your answer to the nearest tenth.
The length of the shadow cast by the 29-m tall building is approximately 13.9 m. This is derived by using the Pythagorean theorem, considering the details of the problem as elements of a right triangle.
Explanation:In this problem, we are given a 29-m tall building and the distance from the top of the building to the tip of the shadow which is 34 m. We are asked to find the length of the shadow. To solve for this, we can use the knowledge of right triangles from geometry.
In this case, the height of the building is one side of a right triangle, the distance from the top of the building to the end of the shadow is the hypotenuse, and the shadow length would be the other side. Since we know the length of the hypotenuse (34 m) and one side (29 m), we can solve for the other side (the shadow length) by using the Pythagorean theorem which states that the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides.
Expressed in the formula: c^2 = a^2 + b^2. Where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.
Rearranging for 'a', which is the length of the shadow, we get: a^2 = c^2 - b^2. Substituting with the given values results in: a = sqrt(34^2 - 29^2). After calculating, the approximate length of the shadow is 13.9 m (rounded to the nearest tenth).
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What are the quotient and remainder of (3x^4+ 2x^2 - 6x + 1) /(x + 1)
Answer:
all work is shown and pictured
A circle with area of 81 pi has a sector with central angle of 120 degrees
Answer:
Find the arc length.
532.95863765
Step-by-step explanation:
Write the coordinates of the vertices after a reflection across the line x= -3
(Please vote me Brainliest if this helped!)
C' (-3, -6)D' (-3, 1)E' (0, -6)A reflection across the line x=-3 is achieved by subtracting each x-coordinate from -6, resulting in a new point of (-6 - x, y). The y-coordinate remains unchanged in the reflection.
Explanation:In mathematics, a reflection is a transformation that uses a line of reflection to create a mirror image of the original figure. In this case, the line of reflection is x = -3.
To reflect a point across this line, we modify only the x-coordinate of each vertex. The rule for a reflection over the line x = -3 is to take each x-coordinate and subtract it from -6 (which is the double of -3). So, if we have a point (x, y), after the reflection the new point would be (-6 - x, y). This results in a new x-coordinate that is the mirror image around the line x = -3, while the y-coordinate remains unchanged.
For example, if we have a vertex at (2, 5), after the reflection, it would be at (-6 - 2, 5), or (-8, 5).
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This regression analysis examines the relationship between the number of years of formal education a person has and their annual income. According to this model, about how much more money do people who finish a 4-year college program earn each year, on average, than those with only a 2-year degree?Dependent variable is Income Variable Coefficient SE Coefficient Constant 3984.45 6600 Education 2668.45 600.1
Answer:
$5336.90
Step-by-step explanation:
First step is to filter through what is given And to deduce the important information....
Kindly go through the attached file for further comprehension and a detailed solution.
To determine the average annual income difference between individuals with a 4-year degree and those with a 2-year degree, multiply the education coefficient (2668.45) by 2, resulting in an approximate difference of $5336.90.
Explanation:Based on the regression analysis provided, the variable coefficient for education is 2668.45. This coefficient represents the average change in annual income associated with an additional year of formal education. If we want to compare the annual earnings of individuals with a 4-year college degree to those with a 2-year degree, we calculate the difference by multiplying the coefficient by the difference in years of education (4 years - 2 years = 2 years).
To find the increase in annual income for those with a 4-year degree compared to a 2-year degree, we perform the following calculation:
2668.45 × 2 = 5336.90.
Therefore, on average, people with a 4-year college degree earn approximately $5336.90 more annually than those with only a 2-year degree, according to this model.
In 2016, the mean score on the AP Art History exam was 3.8, with a standard deviation of 1.2. The mean score on the AP English exam was a 2.9, with a standard deviation of 1.7. What is the combined mean and standard deviation for these exams?
A.) Mean, 0.9; standard deviation, 0.2
B.) Mean, 3.35; standard deviation, 2.9
C.) Mean, 3.35; standard deviation, 1.45
D.) Mean, 6.7; standard deviation, 2.08
E.) Mean, 6.7; standard deviation, 4.33
Answer:
[tex] X+Y \sim N(\mu_X +\mu_Y , \sqrt{\sigma^2_X +\sigma^2_Y})[/tex]
The mean is given by:
[tex] \mu = 3.8+2.9= 6.7[/tex]
And the standard deviation would be:
[tex] \sigma =\sqrt{1.2^2 + 1.7^2} = 2.08[/tex]
And the distribution for X+Y would be:
[tex] X+Y \sim N(6.7 , 2.08) [/tex]
And the best answer would be:
D.) Mean, 6.7; standard deviation, 2.08
Step-by-step explanation:
Let X the random variable who represent the AP Art History exam we know that the distribution for X is given by:
[tex] X \sim N(3.8, 1.2)[/tex]
Let Y the random variable who represent the AP English exam we know that the distribution for X is given by:
[tex] X \sim N(2.9, 1.7)[/tex]
We want to find the distribution for X+Y. Assuming independence between the two distributions we have:
[tex] X+Y \sim N(\mu_X +\mu_Y , \sqrt{\sigma^2_X +\sigma^2_Y})[/tex]
The mean is given by:
[tex] \mu = 3.8+2.9= 6.7[/tex]
And the standard deviation would be:
[tex] \sigma =\sqrt{1.2^2 + 1.7^2} = 2.08[/tex]
And the distribution for X+Y would be:
[tex] X+Y \sim N(6.7 , 2.08) [/tex]
And the best answer would be:
D.) Mean, 6.7; standard deviation, 2.08
To find the combined mean and standard deviation, use weighted average for means and formula for standard deviation of two independent variables.
Explanation:To find the combined mean and standard deviation for the AP Art History and AP English exams, we need to use the formulas for combining means and standard deviations when the two sets of data are independent. The combined mean can be found by taking the weighted average of the two means, where the weights are the number of observations in each dataset.
The combined standard deviation can be found using the formula for the standard deviation of the sum of two independent variables. After calculating, the combined mean is 3.35 and the combined standard deviation is 2.08.
So the correct answer is D) Mean, 3.35; standard deviation, 2.08.
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When trying to find the probability of a white ball being drawn from a bag containing 4 white balls, 3 black balls, and 5 red balls, how many desired outcomes are there?
Answer:
There are 4 desired outcomes.
Step-by-step explanation:
You want to select a white ball from a bag of balls. So the number of desired outcomes is the number of white balls in the bag.
In this problem:
Ball being drawn from a bag containing 4 white balls, 3 black balls, and 5 red balls
4 white balls
So there are 4 desired outcomes.
As predictors become more highly correlated, I. the p-values of the beta estimates become smaller II. it becomes more difficult to determine which predictor is actually producing the effect on the response III. the values of the beta estimates all approach zero
Answer:
Hence P values of beta becomes smaller(< 0.0001). and doest affect the mean response
Step-by-step explanation:
Given:
AS Predictor become more highly correlated .
To find:
Descriptive Nature of high correlated Predictor .
Solution:
A predictor is high correlated means:
1)It means that the two variables are strongly related to each other.
2)This is also called as problem of multicollinearity when two variables are
in Regression.
Effects when predictor are highly correlated ;
The estimated coefficient of one any one variable depends on the other predictor variable in model.Estimated coefficient of regression decrease as predictor variables are added.Hypothesis test Beta = zero gives different conclusion depending upon variable.High correlated of predictor variable does not provide good precision of predication of response in within model.In short ,Mulitcollinearity does not affect the mean response and new response of the model.
Hence P values of beta becomes smaller(< 0.0001). and doest affect the mean response
A certain triangle has its base equal measure to its height. The area of a triangle is 72 square meters. Find the base and height of the triangle.
Answer:
[tex]b = 12\,m[/tex] and [tex]h = 12\,m[/tex]
Step-by-step explanation:
The formula of the area for a triangle is:
[tex]A = \frac{1}{2}\cdot b\cdot h[/tex]
Where:
[tex]b[/tex] - Base
[tex]h[/tex] - Height
But [tex]b = h[/tex], then, the formula is simplified into this form:
[tex]A = \frac{1}{2}\cdot b^{2}[/tex]
Now, all known variables are substituted and the base is:
[tex]b = \sqrt{2\cdot A}[/tex]
[tex]b = \sqrt{2\cdot (72\,m^{2})}[/tex]
[tex]b = \sqrt{144\,m^{2}}[/tex]
[tex]b = 12\,m[/tex]
And the height is:
[tex]h = 12\,m[/tex]
The base and height of the triangle, both having equal measures, are found to be 12 meters each by using the formula for the area of a triangle and quadratic equations.
Explanation:The subject of this problem is triangle geometry within Mathematics. The area of a triangle is calculated using the formula A = 1/2 * base * height. Since we know that the base is equal to the height, we can say that base = height, and hence we can modify the formula to become A = 1/2 * base². Given that the area A is 72 square meters, we substitute this value back into the equation to calculate it as follows:
72 = 1/2 * base².
Solving for base (and consequently, height due to its equal measure), we get base = height = √(72 * 2) = √144 = 12 meters.
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Dayren is driving ti visit family. She drives at an average of 65 miles per hour. She drives 227.5 miles before lunch and then 97.5 miles after lunch. How many hours did she spend driving?
Answer:
5 hours
Step-by-step explanation:
227.5+97.5=325
325/65=5
Find the area of an equilateral triangle (regular 3-gon) with the given measurement: 6-inch radius. A = sq. in.
The correct answer is 27√3
Answer: The area or "the answer" will be 27√3
Step-by-step explanation:
Two bottling plants package a certain type of sports drink. Suppose the mean volume of all of this type of sports drinks is 20 fluid ounces. Bottling plant A bottles an average of 50,000 sports drinks per day. Bottling plant B bottles an average of 175,000 sports drinks per day. On a particular day, which bottling plant is less likely to record a mean volume of 21 fluid ounces for the day?
Answer:
Bottling plant B (with 175,000 sports drinks per day), because the daily mean will be closer to 21 fluid ounces with more sports drinks in the sample.
Given Information:
Mean volume = μ = 20 fl oz
Sampling size of plant A = n₁ = 50,000 drinks/day
Sampling size of plant B = n₂ = 175,000 drinks/day
Required Information:
Which bottling plant is less likely to record a mean volume of 21 fl oz ?
Answer:
Plant B is less likely to record a mean volume of 21 fl oz
Step-by-step explanation:
The standard deviation of the plant A is given by
σa = σ/√n₁
The standard deviation of the plant B is given by
σb = σ/√n₂
Where σ is standard deviation for the mean volume of 20 fl oz and it is fixed.
As you can notice in the above relation, the standard deviation of the plants depend upon the number of samples (n). As the number of sample increases, the standard deviation of samples decreases which means that the mean of the samples will be closer to the actual mean (that is 20 fl oz).
Since the plant B has more samples (n₂ = 175,000) then its standard deviation (σb) will be less and the mean will be closer to 20 fl oz therefore, it is less likely that it will record a mean of 21 fl oz.
Plant A is more likely to record a mean volume of 21 fl oz
Many realistic application involve sampling without replacement. For example, in manufacturing, quality control inspectors sample items from a finite production run without replacement. For such a finite population. we have to adjust the value of sigma(X bar). Take (without replacement) samples of size 2 from the above population of N= 5 objects {1, 2, 3, 4, 5) a. How many such samples are possibleb. List all the Xs, i.e. the means of all these samples of size 2c. Find the mean of these means, i.e. E(X)d. Is E(X)=Mu?e. Find the standard error sigma(X), i.e. the std deviation of all these meansf. Show that the standard error equals Note: is called the finite population correction factor. Typically it is used when the sample size n is greater than 5% of the finite population size.
Answer:
a. possible sample size is 10
b. mean is 3
c standard deviation is 0.9
Step-by-step explanation:
Mean of any given set of numbers is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers.
Standard deviation is a measure of the amount of variation or dispersion of a set of values.
Go to attachment for detailed analysis.
The number of possible samples of size 2 without replacement from a population of 5 objects is 10. The means of these samples can be calculated by taking the combinations of 2 objects and finding their mean. The mean of these means is equal to the population mean. The standard error of these means can be calculated using the variance of the means and the finite population correction factor.
Explanation:a. To calculate the number of possible samples of size 2 without replacement from a population of 5 objects, we use the combination formula: C(n, r) = n! / (r! * (n-r)!), where n is the population size and r is the sample size. In this case, n = 5 and r = 2, so the number of possible samples is C(5, 2) = 5! / (2! * (5-2)!) = 10.
b. To list all the means of these samples, we take each combination of 2 objects from the population and calculate their mean. For example, one possible sample is {1, 2}, and its mean is (1 + 2) / 2 = 1.5. Similarly, we calculate the means for the other 9 possible samples.
c. To find the mean of these means, we calculate the average of all the means calculated in part b. d. The mean of these means, denoted as E(X), is equal to the population mean, denoted as Mu, when the sampling is done without replacement.
e. To find the standard error sigma(X), we need to calculate the standard deviation of all these means. We can do this by calculating the variance of the means and taking its square root. f. The standard error sigma(X) equals the square root of the variance of the means divided by the square root of the sample size, multiplied by the finite population correction factor.
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Kelly’s Experimental Outcomes
Trial
Outcome
1
3, H
2
3, T
3
4, T
4
2, H
5
5, T
Kelly conducted an experiment. In each trial she flipped a coin and rolled a number cube that has sides labeled 1 to 6.
The results are shown above. What is the experimental probability that her next flip will be tails?
Group of answer choices
Answer:
60% or 3/5
Step-by-step explanation:
Tails was the result 3 times.
There were 5 trials.
The probability is 3/5.
Work out the percentage change to 2 decimal places when a price of £60 is increased to £89.99
Answer:
that is the solution to the question
The percentage change to 2 decimal places when a price of £60 is increased to £89.99 is 49.98%.
Given that, a price of £60 is increased to £89.99.
What is percentage?Percentage is defined as a given part or amount in every hundred. It is a fraction with 100 as the denominator and is represented by the symbol "%".
Here, old price =£60 and new price =£89.99
Change in price = 89.99-60
= 29.99
Percentage = Change in price/Old price ×100
= 29.99/60 ×100
= 0.4998 ×100
= 49.98%
Therefore, the percentage change to 2 decimal places when a price of £60 is increased to £89.99 is 49.98%.
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factor 3x^2 +30x + 75
I WILL MARK BRAINLIEST IF ANSWER IS RIGHT
Answer:
3(x+5)^2
Step-by-step explanation:
Step 1:
Extract the obvious common constant factor
3(x^2+10x+25)
Step 2:
Recognize the second term as a square
3(x+5)^2
The selling price for a classic car is $12,000, which is $500 less than two times its original price. What was the original price of the car?
Answer:$6250
Step-by-step explanation:
selling price(sp)=12000
Original price(op)=?
sp+500=2 x op
12000+500=2 x op
12500=2 x op
Divide both sides by 2
12500/2=(2 x op)/2
6250=op
Therefore original price is $6250
To find the original price of the car, set up an equation with the given information and solve for X. The original price of the car was $6,250.
Explanation:
To find the original price of the car, we need to set up an equation based on the given information. Let's assume the original price of the car is X. According to the question, the selling price of the car is $12,000, which is $500 less than two times its original price. So, we can write the equation as:
2X - $500 = $12,000
To solve the equation, we add $500 to both sides:
2X = $12,500
Finally, we divide both sides by 2 to find the original price of the car:
X = $6,250
Therefore, the original price of the car was $6,250.
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Calculate the lower quartile of the data set for the blood glucose levels of 10 individuals whose blood glucose readings were as follows: 88, 97, 101, 104, 104, 107, 109, 117, 121, and 147
Final answer:
To find the lower quartile of a data set, sort the data in ascending order and find the median of the lower half.
Explanation:
To calculate the lower quartile of a data set, you need to find the median of the lower half of the data. In this case, we have 10 blood glucose readings. To find the lower quartile:
First, sort the data in ascending order: 88, 97, 101, 104, 104, 107, 109, 117, 121, 147.
Since we have an even number of values (10), the lower half has 10/2 = 5 values.
The median of the lower half of the data is the average of the two middle values: 104 and 104. So, the lower quartile is 104.
complete the table for the given rule y=x-1/4
Answer:
Is there a picture of the table I will try and solve but I'm not sure if I can
Answer:
0,3,7/4
Step-by-step explanation: In the equation,x-1/4 means 1/4 is subtracted from x to find each value of y,we need to take each value of x and subtract 1/4 for example: when x = 1/4: y = 1/4 - 1/4 , y = 0
What does a dot plot show
Answer:
A Dot Plot, also called a dot chart or strip plot, is a type of simple histogram-like chart used in statistics for relatively small data sets where values fall into a number of discrete bins (categories). ... A dot plot is a graphical display of data using dots
Suppose any one round of a gambling game pays 3 to 1 and the odds are 4 to 1 against you. Suppose you will play for 100 rounds and bet one dollar on each round. We want the probability you will come out ahead.
a. Determine what the possible net gains are. (Not all integers from −100 to 300 are possible.)
b. What is the smallest possible positive net gain?
c. What is the chance you will come out ahead?
Answer:
Check the explanation
Step-by-step explanation:
The odds are 4 to 1 against, so we can estimate the probability of success (p) as
[tex]\frac{p}{q}=\frac{p}{1-p}=\frac{1}{4}\\\\4p=1-p\\\\5p=1\\\\p=0.2[/tex]
The expected pay for every success is 3 to 1, so we lose $1 for every lose and we gain $3 for every win.
The number of winnings in the 100 rounds to be even can be calculated as:
[tex]W+L=100\\\\L=100-W\\\\\\Payoff=0=3*W-1*L=3W-1*(100-W)=3W+W-100\\\\0=4W-100\\\\W=25[/tex]
We have to win at least 25 rounds to have a positive payoff.
As the number of rounds is big, we will approximate the binomial distribution to a normal distribution with parameters:
[tex]\mu=np=100*0.2=20\\\\\ \sigma=\sqrt{npq}=\sqrt{100*0.2*0.8}=4[/tex]
The z-value for x=25 is
[tex]z=\frac{X-\mu}{\sigma}=\frac{25-20}{4}=1.25[/tex]
The probability of z>1.25 is
P(X>25)=P(z>1.25)=0.10565
There is a 10.5% chance of having a positive payoff.
NOTE: if we do all the calculations for the binomial distribution, the chances of having a net payoff are 13.1%.