Answer:
g(5) = 120
Step-by-step explanation:
g(x) = x^3 - 5;
Let x=5
g(5) = 5^3 -5
= 125 -5
= 120
Answer:
g(5) = 120
Step-by-step explanation:
We want to find g(x) when x is equal to 5, so we can substitute 5 in for x.
g(x) = x^3 - 5
g(5) = 5^3 - 5
Solve the exponent
g(5) = 125-5
Subtract
g(5) = 120
A 12 sided die with sides labeled 1 through 12 will be rolled once each number greater than 10?
Answer:
See the explaination
Step-by-step explanation:
Total No. of rolling each side of the 12-sided die= 12 x 12= 144 outcomes
No. of outcomes equals 10 are: (1, 9), (9, 1), (2, 8), (8, 2), (3, 7), (7, 3), (4, 6), (6, 4), (5, 5) = 9 outcomes.
No. of outcomes of rolling a No. >10: 144–9 = 135 outcomes.
Probability, Pr, of rolling a No. >10= 135/144
Pr (No. >10) = 0.9375 (to 4 dec. pl.)
Answer: I think you’re missing something on your question. But I think what your asking is how many numbers greater than 10. It should 2/12 or
Step-by-step explanation:
Linda shoots an arrow at a target in an archery competition. The arc of the arrow can be modeled by the equation y= -0.02x to the power of 2 + 0.65+4 where x is the horizontal distance (in meters) from Linda and y is the height (in meters) of the arrow. How far from Linda does the arrow hit the ground? Round to the nearest tenth.
Answer:
37.8metres
Step-by-step explanation:
The arc of the arrow can be modeled by the equation:
y=-0.02x²+0.65x+4
Where x is the horizontal distance (in meters) from Linda and y is the height (in meters) of the arrow.
The arrow hits the ground when its height (y) is zero.
Therefore, we determine the value(s) of x for which:
y=-0.02x²+0.65x+4=0
Using a calculator to solve the quadratic equation:
x=37.79 or -5.29
Since the distance cannot be a negative value, we ignore -5.29.
The distance from Linda when the arrow hits the ground is 37.8metres (to the nearest tenth)
Answer:
37.8 m
Step-by-step explanation:
Given:-
- The arc trajectory of the arrow is modeled by:
y = -0.02x^2 + 0.65x + 4
Where, x is the horizontal distance (in meters) from Linda
y is the height (in meters) of the arrow
Find:-
How far from Linda does the arrow hit the ground? Round to the nearest tenth.
Solution:-
- We are to determine the range of the projectile trajectory of the arrow. The maximum distance "x_max" occurs when the arrow hits the ground.
- Set the trajectory height of arrow from linda , y = 0:
0 = -0.02x^2 + 0.65x + 4
- Solve the quadratic equation:
x = -5.29 m , x = 37.8 m
- The negative distance x lies at the back of Linda and hence can be ignored. The maximum distance travelled by the arrow would be = 37.8 m
i need the steps for 28
Solve the equation 2x2 + 13x + 15 = 0 by factoring. Show all steps.
Answer:
(2x + 3)(x + 5)
Step-by-step explanation:
2x² + 3x + 15 = 0
(2x + 3)(x + 5)
2x + 3 = 0
2x = -3
x = -3/2
x + 5 = 0
x = -5
In a lab, 180 cells are present at the beginning of an experiment. During the first 9 hours, the number of cells increased by 8% each hour. Write an exponential model giving the number of cells y present t hours after starting the experiment. Estimate the time when the number of cells is 360.
Answer:
The answer is 9 hours
Step-by-step explanation:
From the initial population of 180 cells, the number grows by 8% every hour for the first 9 hours.
This represents a increased population of 108% the previous population.
108% = 108/100 = 1.08
So the new population after n hours is
180 x 1.08^n
When population is 360
360 = 180 x 1.08^n
1.08^n = 2
Take log of both sides
n x Log1.08 = Log 2
n = Log 2/Log 1.08
n = 9
For the given maximization problem, (a) determine the number of slack variables needed, (b) name them, and (c) use slack variables to convert each constraint into a linear equation. a. How many slack variables are needed? 3 b. Which slack variables should be used? A. s 1, s 2, s 3 B. s 1, s 2 C. x 1, x 2, x 3 D. x 1, x 2 c. What is the equation using a slack variable that corresponds to the first constraint, 5 x 1 plus 8 x 2 plus 10 x 3 less than or equals 173? A. 5 x 1 plus 8 x 2 plus 10 x 3 less than or equals 173 B. 5 x 1 plus 8 x 2 plus 10 x 3 plus s 1 less than or equals 173 C. 5 x 1 plus 8 x 2 plus 10 x 3 plus x 1 equals 173 D. 5 x 1 plus 8 x 2 plus 10 x 3 plus s 1 equals 173 What is the equation using a slack variable that corresponds to the second constraint, 5 x 1 plus 4 x 2 plus 17 x 3 less than or equals 245? A. 5 x 1 plus 4 x 2 plus 17 x 3 plus s 2 equals 245 B. 5 x 1 plus 4 x 2 plus 17 x 3 less than or equals 245 C. 5 x 1 plus 4 x 2 plus 17 x 3 plus x 2 equals 245 D. 5 x 1 plus 4 x 2 plus 17 x 3 plus s 2 less than or equals 245 Maximize zequals5x 1plus3x 2plusx 3 subject to: 5 x 1 plus 8 x 2 plus 10 x 3 less than or equals 173 5 x 1 plus 4 x 2 plus 17 x 3 less than or equals 245 with x 1greater than or equals0, x 2greater than or equals0, x 3greater than or equals0
Answer:
a) 2
b) s₁ and s₂
c) First linear equation: 5*x₁ + 8*x₂ + 10*x₃ + s₁ = 173
Second linear equation: 5*x₁ + 4*x₂ + 17*x₃ + s₂ = 254
Step-by-step explanation:
The problem statement, establishes two constraints, each one of them will need a slack variable to become a linear equation, so the answer for question
a) 2.
b) The constraints are: s₁ and s₂
c) First constraint
5*x₁ + 8*x₂ + 10*x₃ ≤ 173
We add slack variable s₁ and the inequality becomes
5*x₁ + 8*x₂ + 10*x₃ + s₁ = 173
The second constraint is:
5*x₁ + 4*x₂ + 17*x₃ ≤ 254
We add slack variable s₂ and the inequality becomes
5*x₁ + 4*x₂ + 17*x₃ + s₂ = 254
Answer:
A. Two slack variables are needed
B. S1 and s2 (option b)
C. Option d (5X1 + 8X2 + 10X3 + s1 = 173)
D. Option a (5X1 + 4X2 + 17X3 + s2 = 245)
E. Z is maximized at 173 when (X1, X2, X3) = (34.6, 0, 0)
Step-by-step explanation:
In a linear maximization problem like this, if we want to convert the inequality (constraint) into a linear equation, we add slack variables to the left hand side of each inequality.
Therefore we add s1 and s2 to the first and second inequality respectively.
[tex]5X1 + 8X2 + 10X3 \leq 173\\5X1 + 4X2 + 17X3 \leq 245[/tex]
Imputing the slack variables, we obtain as follows:
[tex]5X1 + 8X2 + 10X3 + s1 = 173\\5X1 + 4X2 + 17X3 + s2 = 245[/tex]
[tex]Maximize :\\Z = 5X1 + 3X2 + X3[/tex]
Subject to
[tex]5X1 + 8X2 + 10X3 \leq 173\\5X1 + 4X2 + 17X3 \leq 245\\With \\X1 \geq 0\\X2 \geq 0\\X3 > 0[/tex]
Solution :
[tex]5X1 + 8X2 + 10X3 + s1 = 173\\5X1 + 4X2 + 17X3 + s2 = 245\\-5X1 - 3X2 - X3 + Z = 0[/tex]
5X1 on the first equation is the pivot because the negative value of -5 is the highest and 173/5 is less than 245/5
We perform eq1 - eq2 on the first row and eq1 + eq3 on the third row to get X1 and clear the rest.
We
X1 = 173/5 = 34.6
X2 = 0 (inactive)
X3 = 0 (inactive)
s1 = 0 , s2 = -72/-1 = 72,
Z = 173/1 =173
Therefore the minimum value of Z is 173 when (X1, X2, X3) = (34.6, 0, 0)
The dinner buffet offers a choice of 4 appetizers, 5 main courses and 4 desserts. How many possible appetizer-main course-dessert combinations are there? *
1 point
Step-by-step explanation:
The dinner buffet offers a choice of appetizers,main courses and desserts.
The number of appetizers available in the buffet = 4
The number of main courses available in the buffet = 5
The number of desserts available in the buffet = 4
By combinations, the possible appetizer-main course-dessert combinations are there in the dinner buffet = (4) (5) (4) = 80
Which of the following are in the correct order from least to greatest? StartFraction 3 pi Over 10 EndFraction, 60°, StartFraction pi Over 2 EndFraction, StartFraction 2 pi Over 3 EndFraction, 255° 255°, StartFraction 2 pi Over 3 EndFraction, StartFraction pi Over 2 EndFraction, 60°, StartFraction 3 pi Over 10 EndFraction StartFraction 3 pi Over 10 EndFraction, StartFraction pi Over 2 EndFraction,StartFraction 2 pi Over 3 EndFraction 60°, 255° 255°, 60°, StartFraction 2 pi Over 3 EndFraction, StartFraction pi Over 2 EndFraction, StartFraction 3 pi Over 10 EndFraction
Answer:
D) 69.1 ft
Step-by-step explanation:
Answer: a
Step-by-step explanation:
what is 2(x + 8) = x + 21
Answer:
x=5
Step-by-step explanation:
2(x + 8) = x + 21
Distribute
2x+16 = x+21
Subtract x from each side
2x-x +16 = x-x+21
x +16 = 21
Subtract 16 from each side
x+16-16 =21-16
x =5
The triangular prism below has a base area of 9 cm and a height of 23 cm. What is the volume of the prism?
A) 207 cm3
B) 103.5 cm3
C) 1863 cm3
D) 200 cm3
Answer:
A) 207 cm3
Step-by-step explanation:
9 x 23 = 207
A recent study investigated the extent to which hypertension is affected by alcohol consumption. Among other things, the study compared the rate of hypertension in nondrinkers to the rate in people that are moderate beer drinkers (average between 0.5 and 1 glass of beer per day). They found that a 95% confidence interval for the relative risk of hypertension in moderate beer drinkers as compared to nondrinkers was 0.47 to 0.90. A relative risk of 1 means that the two groups have the same risk of hypertension, while a relative risk of less than 1 means that moderate beer drinkers have less risk of hypertension than nondrinkers. If the researchers who ran the study had tested the hypotheses.H0: the relative risk for moderate beer drinkers as compared to nondrinkers = 1
Ha: the relative risk for moderate beer drinkers as compared to nondrinkers ≠ 1
they would have concluded which of the following?
a) reject H0
b) fail to reject H0 at level 0.10
c) Neither choice is correct. We cannot determine what we would do at level a = 0.10 because the interval given is a 95% confidence interval, not a 90% confidence interval.
Answer:
a) reject H0
Step-by-step explanation:
The null and alternative hypotheses:
H0: the relative risk for moderate beer drinkers as compared to nondrinkers = 1
Ha: the relative risk for moderate beer drinkers as compared to nondrinkers ≠ 1
The level of significance, α = 0.1
Since the confidence interval for the relative risk of hypertension in moderate beer drinkers as compared to nondrinkers (0.47 to 0.90) does not contain the relative risk of 1, the null hypothesis will be rejected at the specified level of significance, 0.1. Therefore, the conclusion is to reject H0.
What is the area of a triangle that has a base of 15 yards and a height of 11yards
Answer:
6
Step-by-step explanation:
you have to multiply it by joe mama
A bank wishes to estimate the mean balances owed by customers holding MasterCard. The population standard deviation is estimated to be $300. If a 98 percent confidence interval is used and an interval of $75 is desired, how many cardholders should be sampled?a. 212
b. 87
c. 629
d. 44
Answer:
B. 87
Step-by-step explanation:
The first thing is to calculate critical z factor
the alpha and the critical z score for a confidence level of 90% is calculated as follows:
two sided alpha = (100% - 98%) / 200 = 0.01
critical z factor for two sided alpha of .01 is calculated as follows:
critical z factor = z factor for (1 - .01) = z factor for (.99) which through the attached graph becomes:
critical z factor = 2.33
Now we have the following formula:
ME = z * (sd / sqrt (N) ^ (1/2))
where ME is the margin of error and is equal to 75, sd is the standard deviation which is 300 and the value of z is 2.33
N the sample size and we want to know it, replacing:
75 = 2.33 * (300 / (N) ^ (1/2))
solving for N we have:
N = (2.33 * 300/75) ^ 2
N = 86.86
Which means that the sample size was 87.
a campground owner had 150 meters of fencing to enclose a rectangualr field adjacent to a river and also bisect that field into two equal halves. No fencing required with river. what are the dimesions of the largest area that can be enclosed
Answer:
Dimension of the Rectangle : 2W - 150 by W
Step-by-step explanation:
Given
Land Size = 150 metres
This land size represents the perimeter of the land
Let P = Perimeter
So, P = 150 m
To bisect the field into two equal halves means to divide length or width of the field into two equal halves
Let L = Length of the field
Let W = Width of the field
Assuming the length of the field is to be bisected., then the campground owner is left with 0.5L (i,e half of the length)
Perimeter of a rectangle is calculated by: 2(Length + Width)
P = 2(L + W) by substitution, becomes
150 = 2(0.5L + W) ----open bracket
150 = L + 2W ---- Male L the subject the formula
L = 2W - 150
The dimension of the largest area that can be enclosed id given by (L,W)
By substitution; the expression becomes
Dimension of the Rectangle : 2W - 150 by W
26 - 2x
3 x - 4
What is the value of x?
Answer:
x = 6
Step-by-step explanation:Well if there's one value of x that fits both of them your first step is to set them equal to each other.
26 - 2x = 3x - 4
Now lets solve it
26 - 2x = 3x - 4
+2x +2x
26 = 5x - 4
+4 +4
30 = 5x
30/5 = 5x/5
6 = x
Gary earns $42,990 per year he is paid weekly he currently has a $456 per month car loan payment and he pays $1277 per year for auto insurance is one weeks paycheck enough to pay for his monthly auto loan and his monthly cost of insurance? Explain
Answer:
Yes
Step-by-step explanation:
There are about 52 weeks in a year, 42,990 ÷ 52 = 826.73076 ,
1277 ÷ 12 = 106.41666
826.73076 - 106.41666 = 720.3141
720.3141 - 456 = 264.3141
Yes, one week's paycheck is enough to pay for his monthly auto loan and his monthly cost of insurance.
In mathematics, it deals with numbers of operations according to the statements.
There are 52 weeks in a year,
Income of 1 week = $42,990/52 = 826.73
So his income of Gary for 1 week is $826.73
he pays $1277 per year for auto insurance
There are 12 months in a calender year
Monthly auto insurance pay = 1277/12 = 106.42
Pay for monthly auto insurance is $106.42
and he currently has a $456 per month car loan payment
Total pay = 106.42 + 456 = $562.42
Here, their income of Gary for a week is greater than the total pay,
Thus, yes, one week's paycheck is enough to pay for his monthly auto loan and his monthly cost of insurance.
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What is the value of the soccer ball?
1. baseball + basketball = 240
2. tennisball + soccerball = 500
3. soccerball - basketball + tennisball = 455
4. baseball + tennisball = 215
Subtract 1 from 4
5. tennisball - basketball = -25
Rearrange 3
6. soccerball + tennisball - basketball = 455
Substitute 5 into 6
7. soccerball - 25 = 455
Solve
soccerball = 480
A box is being created out of a 15 inch by 10 inch sheet of metal. Equal-sized squares are cutout of the corners, then the sides are folded up to create an open-top box. What size squaresshould be cut out so that the box has the largest possible volume?
Answer:
Therefore the dimensions of the square should be 0.1528 inch by 0.1528 inch so, the box has largest volume.
Step-by-step explanation:
Given that,
A box is being created out of a 15 inches by 10 inches sheet of metal.
The length of the one side of the squares which are cut out of the each corners of the metal sheet be x.
The length of the metal box be = (15-2x) inches.
The width of the metal box be =(10-2x) inches
The height of the metal box be =x inches
Then, the volume of the metal box= length×width×height
=(15-2x)(10-2x)x cubic inches
=(150x-50x²+4x³) cubic inches
∴ V= 4x³-50x²+15x
Differentiating with respect to x
V'=12x²-100x+15
Again differentiating with respect to x
V''=24x-100
For maximum or minimum value, V'=0
12x²-100x+15=0
Apply quadratic formula [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex], here a=12, b= -100 and c=15
[tex]x=\frac{-(-100)\pm\sqrt{(-100)^2-4.12.15}}{2.12}[/tex]
[tex]\Rightarrow x=\frac{100\pm\sqrt{9280}}{2.12}[/tex]
[tex]\Rightarrow x=0.1528,8.18[/tex]
For x= 8.18, The value of (15-2x) and (10-2x) will negative.
∴x=0.1528 .
Now, [tex]V''|_{x=0.1528}=24(0.1528)-100<0[/tex]
∴At x=0.1528 inch, the volume of the metal box will be maximum.
Therefore the dimensions of the square should be 0.1528 inch by 0.1528 inch so, the box has largest volume.
The sculpture ‘Cubo Vazado’ [Emptied Cube] by the Brazilian artist Franz Weissmann is formed by removing cubical blocks from a solid cube to leave the symmetrical shape shown.
If all the edges have length 1, 2 or 3, what is the volume of the sculpture?
Answer:
The volume of the sculpture is 12 cubic units
Step-by-step explanation:
The picture of the question in the attached figure
step 1
Find the volume of the L-shaped figure
The volume is given by
[tex]V=Bh[/tex]
where
B is the area of the base
h is the height of the figure
we have
[tex]h=1\ units[/tex]
The area of the base B is equal to the area of the complete square (3 units by 3 units) minus the area of the interior square (2 units by 2 units)
[tex]B=3^2-2^2=5\ units^2[/tex]
so the volume of the L-shaped figure is equal to
[tex]V=(5)(1)=5\ units^3[/tex]
step 2
Find the volume of the sculpture
we know that
The volume of the sculpture is equal to the volume of the L-shaped figure, multiplied by two plus the volume of two unit cubes
so
[tex]V=2(5)+2(1)=12\ units^3[/tex]
If half of the pie below is eaten, what fraction remains?
Answer:
u have the answer tho right
Step-by-step explanation:
In mathematical terms, a whole pie is considered as 1 (or 1/1). If half (1/2) of it is eaten, what remains is the other half (1/2). So, the fraction that remains of the pie is 1/2.
Explanation:When we talk about a pie in mathematics, we usually refer to it as a whole. If you've eaten half of the pie, essentially, you've consumed 1/2 of the pie. If that amount is subtracted from the whole pie (which is always considered as 1 or 1/1), what remains is the other half of the pie. Meaning, if you subtract 1/2 from 1 (or 1/1), you're left with 1/2 of the pie. Hence, if half of the pie is eaten, the fraction that remains is 1/2.
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Can you help me with this please
Step-by-step explanation:
This is a cube
Formula surface area of a cube = s2 × 6
Where s is one side
Square of 8 = 64
64 × 6 = 384
A pizza is 14 inches in diameter. Each square inch of pizza has 14.04 calories. If each slice contains about 270 calories, how many slices is the pizza cut into?
Answer:
8 slices
Step-by-step explanation:
To answer this question, we first need to know the area covered by the pizza.
Since it is circular in shape, we use the formula for the area of a circle.
Mathematically area of a circle = pi * r^2
Here, we were given the diameter but we know that D = 2r or r = D/2. This shows that the radius would be 14/2 = 7 inches
Now the area of the pizza is 22/7 * 7 * 7 = 22 * 7 = 154 square inch
Now we proceed to calculate the amount in calories present in the pizza;
That would be 154 * 14.04 = 2162.16 calories
one slice has 270 calories; The number of slices in 2162.16 calories would be 2162.16/270 = 8.008 which is approximately 8 slices
WILL GIVE BRAINLIEST!
Answer:A D E
Step-by-step explanation:
Answer:
Can i have breainliest.
what does 678×25 equal
Answer:
16950
Step-by-step explanation:
Answer:
16950
PLEASE MARK BRAINLIEST IT TOOK A LING TIME TO FIGURE OUT!! THANK YOU!!
Which expressions are equivalent to 5 x minus 15? Select three options. 5 (x + 15) 5 (x minus 3) 4 x + 3 y minus 15 minus 3 y + x Negative 7 y minus 6 x minus 8 y + x Negative 20 minus 3 x + 5 + 8 x
Answer:
B) 5 (x minus 3)
C) 4 x + 3 y minus 15 minus 3 y + x
E) Negative 20 minus 3 x + 5 + 8 x
tep-by-step explanation:
5x minus 15
5x - 15
5(x - 3)
4 x + 3 y minus 15 minus 3 y + x
4x + 3y - 15 - 3y + x
5x - 15
Negative 20 minus 3 x + 5 + 8 x
-20 - 3x + 5 + 8x
5x - 15
Answer:
answers are b,c,e
Step-by-step explanation:
There are 5 cups of oatmeal in a container. Jessica eats 1/3 cup of the oatmeal every day for breakfast. In how many days will Jessica finish all the oatmeal in the container?
Answer:
she will finish the oatmeal in 15 days
Step-by-step explanation:
it takes 3 days to eat one cup. 3x5= 15.
hope this helps. :)
Answer:
It would take Jessica 15 days to finish the container.
Step-by-step explanation:
To work this out you would first multiply 1/3 by 3, which gives you 3/3 or 1. This shows that it would take Jessica 3 days to eat 1 cup of oatmeal. Then you would multiply 3 by 5, which is 15. This means that it would take Jessica 15 days to finish the container.
1) Multiply 1/3 by 3.
[tex]\frac{1}{3} *3=\frac{3}{3} or 1[/tex]
2) Multiply 3 by 5.
[tex]3*5=15[/tex]
Write equation of a line
Parallel to y=2x+3 and passes through point (-2,-1)
Answer:
The answer to your question is y = 2x + 1
Step-by-step explanation:
Data
line y = 2x + 3
point (-2, -1)
line parallel = ?
Process
1.- Find the slope of the original line.
The slope is the coefficient of the x, then the slope is 2
y = 2x + 3
2.- Find the equation of the parallel line using the point slope equation
y - y1 = m(x - x1)
-Substitution
y + 1 = 2(x + 2)
-Simplification
y + 1 = 2x + 2
y = 2x + 2 - 1
-Result
y = 2x + 1
Note: Figure is not drawn to scale.
If r = 10 units and x = 13 units, then what is the volume of the cylinder shown above?
V cylinder =
2h
O A.
460 T cubic units
OB. 130 cubic units
oc. 1,300 cubic units
OD. 5,200 cubic units
Final answer:
To find the volume of the cylinder with a given radius of 10 units and height of 13 units, we use the formula V = πr²h, which yields an approximate volume of 4085.48 cubic units using π as 3.1416. The nearest given option is C (1300 cubic units), although the calculated volume does not exactly match any of the provided choices.
Explanation:
To calculate the volume of a cylinder, we use the formula V = πr²h, where V represents the volume, r is the radius, and h is the height of the cylinder. Given the radius r as 10 units and the height x (which likely represents the height h in this context) as 13 units, we plug these values into the formula:
V = π(10²)(13) = π(100)(13) = 1300π cubic units.
Since π is approximately 3.1416, the volume is approximately:
V ≈ 1300(3.1416) ≈ 4085.48 cubic units.
As none of the multiple-choice options provided exactly match this calculated volume, this could indicate a typo or error in the question or options. However, if we use a rounded value of π as 3.14, then:
V ≈ 1300(3.14) ≈ 4082 cubic units.
The closest match to this calculated volume is option C (1300 cubic units). However, the student should verify the precision of π used in their classroom or the context of the question to ensure this match is consistent with their requirements.
A jar contains 6 blue cubes, 7 blue spheres, 3 green cubes, and 4 green spheres. If you select an object at random, what is the probability that the object is green or a cube?
A. 4
B. 13/20
C. 4/5
D. 3/20
In a game of roulette, Jorge places 170 bets of each on the number 3. A win pays off with odds 35:1 and on any one spin there is a probability that 3 will be the winning number. Among the 170 bets, what is the minimum number of wins needed for Jorge to make a profit? Estimate the probability that Jorge will make a profit.
Answer:
An estimate of the probability that Jorge will make a profit is (5; 0.496)
Step-by-step explanation:
The total cost = 170x1 = $170
The payoff is $35 per $1 bet
The number of wins needed to make a profit = 170/35 = 4.86 \approx 5
Probability of winning, P(win), p = 1/38
n = 170
P(Jorge will make a profit) = P(at least 5 wins)
mean = np = 4.47
standard deviation = \sqrt{npq} = 2.09
P(X \geq 5) = 1 - P(X < 5)
P(X < A) = 1 - P(Z < (A - mean)/standard deviation)
After the application of continuity correction,
P(X \geq 5) = 1 - P(Z < (4.5 - 4.47)/2.09)
= 1 - P(Z < 0.01)
= 1 - 0.5040
P(X \geq 5 = 0.496
An estimate of the probability that Jorge will make a profit is (5; 0.496)
Answer:
probability that Jorge makes a profit is = 0.46412
Step-by-step explanation:
Solution:-
- The number of bets made on number "3", N = 170
- He bets on each number "3", k = $1
- The winning pay-off odds : $ ( 35 : 1 )
- The probability of getting number "3" on a spin, p = 1/38
- The total amount paid (C) for n = 170 bets on number "3" are:
C = N*k
C = (170)*($1)
C = $170
- The probability of getting a number "3" on a spin is independent for each trial.
Denote:
- The amount received per win = $ 35
- The number of wins = r
- So the minimum "N" number of wins must be enough to match loss.
Amount Win = Amount Loss
r*$35 = C
r*$36 = C
r = $170 / 36
r = 4.7222 ≈ 5 wins
- So the minimum amount of wins required by r = 10 to make a profit.
- Let a random variable "X" denote the number of times Jorge spins to get number "3" - Number of wins. The probability to get a number "3" on each spin is independent for each trial. Therefore X follows Binomial distribution.
- So, X ~ B ( N , p )
X ~ B ( 170 , 1/38 )
1 - p = 37 / 38
- So we need to determine that Jorge get number "3" at-least r = 5 times. Where the probability mass function for binomial distribution is given below:
[tex]P ( X = r ) = ^NC_r * (p)^r * ( 1 - p )^(^N^ -^ r^ )[/tex]
So,
[tex]P ( X \geq 5 ) = 1 - P ( X \leq 4) = 1 - [ P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) + P ( X = 3 ) + P ( X = 4 )]\\\\1 - [ (37/38)^1^7^0 + 170*(1/38)*(37/38)^1^6^9 + 170C2*(1/38)^2*(37/38)^1^6^8 + \\\\170C3*(1/38)^3*(37/38)^1^6^7 + 170C4*(1/38)^4*(37/38)^1^6^6 ]\\\\1 - [ 0.01074 + 0.04935 + 0.11271 + 0.17059 + 0.19249]\\\\= 1 - 0.53588\\\\= 0.46412[/tex]
- So the probability that Jorge makes a profit is = 0.46412
Note:- The normal approximation to Binomial distribution may be a less cumbersome choice; however, care must be taken to verify the conditions for normal approximation i.e
N*p ≥ 10
With the given data, N = 170 , p = 1/38:
N*p = 170/38 = 4.4737 ≤ 10
Hence, the normal approximation is an invalid choice for the data given.