Answer: the answer is 8:1
Step-by-step explanation:
48:6/6
8:1
for a group of objects made of the same material the weight of an object varies directly with its volume.If an object that has a volume of 30 cubic inches weighs 24 ouncess what is the constant of variation
If an object that has a volume of 30 cubic inches weighs 24 ouncess then the constant of variation is 0.8.
If the weight of an object varies directly with its volume, then we can write the following proportion
Weight / volume = constant of variation
Let's use w to represent the weight of the object and v to represent its volume. We are given that an object with a volume of 30 cubic inches weighs 24 ounces, so we can write
w / v = k
Where k is the constant of variation. We can solve for k by substituting the given values
24 / 30 = k
Simplifying this expression, we get
0.8 = k
Therefore, the constant of variation is 0.8.
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The value of a computer that originally sold for $800 can be modeled by g(x)=800(0.8^x), where x is the number of years since the computer was purchased. Describe g(x) as a translation of the parent function f(x)=0.8^x.
Answer:
Step-by-step explanation:
The parent function is f(x) = 0.8^x, which represents exponential decay. This means that as x increases, f(x) gets smaller and smaller.
The given function g(x) is obtained by multiplying f(x) by 800, which stretches the graph vertically by a factor of 800.
Also, the function g(x) has an additional translation: the entire graph of f(x) is shifted vertically upwards by 800 units. This is because the computer originally sold for $800, and g(x) represents the current value of the computer, which is $800 times the value of the parent function.
Therefore, g(x) can be described as a vertical stretch and translation of the parent function f(x) = 0.8^x.
what is the answer to 2x0.5x9
Answer:
9
Step-by-step explanation:
Alexi's restaurant bill is $58, and he wants to leave a 20 percent tip. Which expression represents the total amount that
Alexi needs to pay?
$58(0.20) + $58
$58(0.20)
$58(20) + $58
$58(20)
Answer:
$58(0.20) + $58
Step-by-step explanation:
this is true because you would need to find what 20% of 58 is before you add it to the bill
Complete the table.
(Type an integer or a simplified fraction.)
please help, no links
The complete table is as follows
x y
0 -11/2
1 -3
2 -1/2
3 2
4 9/2
5 7
How to complete the tableThe table is completed by identifying the constant change
-11/2 + k = -3
k = -3 + 11/2
k = 5/2
This is used to complete the table as follows
For x = 1
-11/2 + 5/2 = -3
For x = 2
-3 + 5/2 = -1/2
For x = 3
-1/2 + 5/2 = 2
For x = 4
2 + 5/2 = 9/2
For x = 5
9/2 + 5/2 = 7
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What fraction does the red dot represent on the number line above?
Answer:
The answer would be Option C. [tex]\frac{1}{8}[/tex]
Step-by-step explanation:
List the sample space for rolling a fair eight-sided die.
S = {1}
S = {8}
S = {1, 2, 3, 4, 5, 6}
S = {1, 2, 3, 4, 5, 6, 7, 8}
The sample space for rolling a fair eight-sided die is S = {1, 2, 3, 4, 5, 6, 7, 8}.
The sample space is the set having the total outcomes of a random trial.
So, in a fair eight-sided die, the outcomes will be =
1, 2, 3, 4, 5, 6, 7, 8 = 8
Writing in the of set, S = {1, 2, 3, 4, 5, 6, 7, 8}.
Hence the sample space for rolling a fair eight-sided die is S = {1, 2, 3, 4, 5, 6, 7, 8}.
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Jensine earns $6 per hour babysitting,
She is saving to buy an MP3 player
that costs $150, but she does not yet
have enough money. If h represents
the number of hours she has spent
babysitting, which inequality describes
her situation?
507
Answer:
well maybe if Jensine gets a better f ***** job she can buy it instead she wants to waste her time to watch a annoying a *baby for 6$ a hour
Step-by-step explanation:
This is all a joke dont get mad :)
the tennis team wins 73% of their games. if they play 10 games, what is the probability that they win nine games?
The Probability that the tennis team won 9 games is 1.35% , under the given condition that the tennis team wins 73% of their game, if they play 10 games.
Therefore, probability of winning nine games out of ten with a 73% win rate can be evaluated using the binomial distribution formula
[tex]P(X=k) = (n choose k) * p^k * (1-p)^{(n-k)}[/tex]
Here
P(X=k) = probability of winning k games out of n games
n = total number of games played (n=10)
k = number of games won (k=9)
p = probability of winning a single game (p=0.73)
Staging the values in the formula
[tex]P(X=9) = (10 choose 9) * 0.73^9 * (1-0.73)^{(10-9)}[/tex]
= ( 10 - 9) x (0.05) x ( 0.27)¹
= 1 x 0.05 x 0.27
= 0.0135
Converting into percentage
0.0135 x 100
= 1.35%
The probability that the tennis team won 9 games is 1.35% , under the given condition that the tennis team wins 73% of their game, if they play 10 games.
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(7, –16)
(8, 0)
(7, 5)
Is this relation a function
The given relation (7, –16) (8, 0) (7, 5) is not a function.
To determine if the given relation is a function, we need to check if each input (x-value) is associated with only one output (y-value).
If there exists an input value that is associated with multiple output values, then the relation is not a function.
Looking at the given relation, we have:
(7, –16)
(8, 0)
(7, 5)
We can see that the input value 7 is associated with two different output values (-16 and 5).
When a relation is a function, each input value is associated with only one output value. In other words, each x-value has only one corresponding y-value. In this case, we have an input value of 7 that is associated with two different output values, which violates the definition of a function.
Therefore, the given relation is not a function.
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A. Write true or false after each sentence. If the sentence
is false, change the underlined word or words to make
it true.
1. The order of the numbers in an ordered pair is not important.
2. The horizontal axis is called the x-axis.
3. The point where the coordinate axes cross is called the center.
1. True
2. True
3. False. The point where the coordinate axes cross is called the origin.
(Comparing Data MC)
The box plots display measures from data collected when 20 people were asked about their wait time at a drive-thru restaurant window.
A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 10 to 14.5 on the number line. A line in the box is at 12.5. The lines outside the box end at 5 and 20. The graph is titled Fast Chicken.
A horizontal line starting at 0, with tick marks every one-half unit up to 32. The line is labeled Wait Time In Minutes. The box extends from 8.5 to 15.5 on the number line. A line in the box is at 12. The lines outside the box end at 3 and 27. The graph is titled Super Fast Food.
Which drive-thru is able to estimate their wait time more consistently, and why?
Fast Chicken, because it has a smaller IQR
Fast Chicken, because it has a smaller range
Super Fast Food, because it has a smaller IQR
Super Fast Food, because it has a smaller range
From the given question we cannot conclude that Burger Quick has a smaller range based on the given information.
Range refers to the set of potential target value, and if we have a graph, then the range is the collection of all feasible y-values on the graph.
Burger Quick is able to estimate their wait time more consistently because it has a smaller interquartile range (IQR). The IQR is a measure of the spread of the middle 50% of the data, and a smaller IQR indicates that the data is more tightly clustered around the median. In this case, Burger Quick has an IQR of 14.5 (from 9.5 to 24), while Fast Chicken has an IQR of 4.5 (from 10 to 14.5). This means that the wait times reported by Burger Quick are more consistent and less variable than those reported by Fast Chicken.
The range is a measure of the total spread of the data, and while a smaller range can indicate less variability, it doesn't necessarily mean that the data is more consistent in the middle. Therefore, we cannot conclude that Burger Quick has a smaller range based on the given information.
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Determine the number of solutions to the given system. Remember to use the format (x,y) to type in your points, and do not use spaces. If there is only one, type the point in one space and "none" in the other. If there is none, type none in both the boxes. System: {y=−x+10y=(x−5)2+6
The two solutions to the system as a whole of equations is (3, 7) as (7, 3), respectively. These locations show where a line [tex]y = x+10[/tex] with the parabola of [tex]y = (x^{5} )^{2+6}[/tex] cross.
We can change the initial equation in the second to solve a set of equations[tex]y=x+10[/tex], [tex]y = (x^{5} )^{2+6}[/tex] , yielding: [tex]−x+10 = (x^{5} )^{2+6}[/tex]
When we simplify and enlarge the right side, we obtain: [tex]x^{2} - 11x + 21 = 0[/tex]The quadratic equation is factored to yield: (x - 3)(x - 7) = 0.
x has solutions of 3 and 7. In order to determine the associated y-values, we may then enter these values back into one of the original equations. There are two possible answers: (3, 7) and (7, 3).
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Computing inverse laplace transforms. Determine the function of time, x(t), for each of the following laplace transforms
1/(s2 + 9), Re {s} >0
The function of time x(t) corresponding to the Laplace transform 1/(s^2 + 9), Re{s} > 0, is x(t) = (1/3π) sin(3t) e^(st).
To compute the inverse Laplace, transform of 1/(s^2 + 9), we can use the formula for the inverse Laplace transform of a rational function:
L^-1{F(s)} = (1/2πi) ∫γ+σ-iγ+σ+ iF(s)e^(st) ds
where γ is a real number greater than the real part of all singularities of F(s), σ is a positive real number such that the contour of integration lies to the right of all singularities of F(s), and the contour of integration γ+σ is a line parallel to the imaginary axis.
In this case, the Laplace transform of 1/(s^2 + 9) is:
F(s) = L{1/(s^2 + 9)} = 1/[(s + 3i)(s - 3i)]
which has singularities at s = ±3i. Since Re{s} > 0, we can choose γ = 0 and σ > 3. Then, the inverse Laplace transform of F(s) is:
L^-1{F(s)} = (1/2πi) ∫γ+σ-iγ+σ+ iF(s)e^(st) ds
= (1/2πi) ∫γ+σ-iγ+σ+ i [1/((s + 3i)(s - 3i))] e^(st) ds
We can use partial fraction decomposition to express F(s) as:
F(s) = A/(s + 3i) + B/(s - 3i)
where A = 1/(2(3i)), B = -1/(2(3i)), and we get:
L^-1{F(s)} = (1/2πi) [∫γ+σ-iγ+σ+ i A/(s + 3i) e^(st) ds + ∫γ+σ-iγ+σ+ i B/(s - 3i) e^(st) ds]
= (1/2πi) [A e^(-3it) ∫γ+σ-iγ+σ+ i e^(su) du + B e^(3it) ∫γ+σ-iγ+σ+ i e^(sv) dv]
= (1/2πi) [(A e^(-3it) + B e^(3it)) ∫γ+σ-iγ+σ+ i e^(su) du]
where u = s - 3i, v = s + 3i, and we can evaluate the integral using the residue theorem:
∫γ+σ-iγ+σ+ i e^(su) du = 2πi Res[e^(su)/(u + 3i), u = -3i]
= 2πi e^(-3it)/(2(3i))
= -i/3 e^(-3it)
Therefore, we have:
x(t) = L^-1{F(s)} = (1/2πi) [(A e^(-3it) + B e^(3it)) ∫γ+σ-iγ+σ+ i e^(su) du]
= (1/2πi) [(1/(2(3i)) e^(-3it) - 1/(2(3i)) e^(3it)) (-i/3) e^(st) ds]
= (1/6π) [e^(-3it) - e^(3it)] e^(st) ds
= (1/3π) sin(3t) e^(st)
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managers should never make decisions unless they have enough data to make a perfect statistical decision based on sufficient values for the mean, standard deviation, and sample size. managers should never make decisions unless they have enough data to make a perfect statistical decision based on sufficient values for the mean, standard deviation, and sample size. true. managers who fail to account for all possible data, even if a decision must be delayed, will always make an inferior decision. true. managers who take the time to consult with their staff and key sources of evidence will never arrive at an inferior decision. false. managers can usually just have a mean difference and make a judgment call that is right at a 95% confidence interval. false. although math and data can help make decisions clearer, eventually we always just have to make a judgment call.
Mathematics and data can help make decisions clearer, but ultimately you only have to make one decision at a time. This statement is true.
Information and factual investigation give profitable bits of knowledge and back decision-making, but it isn't continuously conceivable or fundamental to have full factual information sometime recently making decisions.
Decisions often have to be made with fragmented or fragmented data, and managers use them to possess judgment and encounter to weigh accessible alternatives, and make the most excellent conceivable choices.
A decision must be made. Additionally, decisions often involve more than purely numeric data, such as B.
Social and ethical considerations require thoughtful and nuanced decisions that are not always fully captured by statistical analysis.
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The line plots represent data collected on the travel times to school from two groups of 15 students.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 4, 6, 14, and 28. There are two dots above 10, 12, 18, and 22. There are three dots above 16. The graph is titled Bus 47 Travel Times.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 10, 16, 20, and 28. There are two dots above 8 and 14. There are three dots above 18. There are four dots above 12. The graph is titled Bus 14 Travel Times.
Compare the data and use the correct measure of center to determine which bus typically has the faster travel time. Round your answer to the nearest whole number, if necessary, and explain your answer.
Bus 47, with a median of 16
Bus 14, with a median of 14
Bus 47, with a mean of 16
Bus 14, with a mean of 14
The best measure of center to determine which bus typically has the faster travel time is: A. Bus 47, with a median of 16.
What is a line plot?In Mathematics and Statistics, a line plot simply refers a type of graph that is used for the graphical representation of data set above a number line, while using crosses, dots, or any other mathematical symbol.
For Bus 47, the median is given by;
4,6,14,28,10,10,12,12,18,18,22,22,16,16,16
Median of Bus 47 = 16
Mean of Bus 47 = 15.
For Bus 14, the median is given by;
10,16,20,28,8,8,14,14,18,18,18
Median of Bus 14 = 16.
Mean of Bus 14 = 16
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Find the excluded values
Answer:
[tex]2 {a}^{2} - 8a = 0[/tex]
[tex]2a(a - 4) = 0[/tex]
[tex]a = 0[/tex]
[tex]a = 4[/tex]
The excluded values are a = 0 and a = 4.
4. Allie orders candles from an online company that offers a flat rate for shipping. She placed an order for 4 candles
for $35. A few months later, she placed an order for 12 candles for $49.
a. Define your Variables
b. What information were you given?
c. Create an equation using the information you were given.
d. How much was the shopping?
e. How many candles can she get for $60?
a. Let x be the cost of each candle, and let s be the flat rate shipping fee.
b. Allie placed two orders: one for 4 candles at a cost of $35, and one for 12 candles at a cost of $49.
c. From the first order, we know that:
4x + s = 35
From the second order, we know that:
12x + s = 49
d. To find the shipping fee, we can subtract the cost of the candles from the total cost of each order. For the first order:
s = 35 - 4x
Substituting this into the second equation, we get:
12x + (35 - 4x) = 49
Simplifying and solving for x, we get:
8x = 14
x = 1.75
Substituting this value of x into the first equation to solve for s, we get:
4(1.75) + s = 35
s = 28 - 4(1.75) = 21
Therefore, the shipping fee was $21.
e. Let n be the number of candles she can get for $60. We can set up an equation based on the cost per candle:
nx + s = 60
Substituting the values we found for x and s, we get:
n(1.75) + 21 = 60
Solving for n, we get:
n = (60 - 21) / 1.75 = 22.29
Since Allie cannot buy a fraction of a candle, she can buy a maximum of 22 candles for $60.
Please help me with this math!!
Answer:
Step-by-step explanation:
Distance formula looks a little like pythagorean because that where it comes from
d = [tex]\sqrt{(x2-x1)^{2}+(y2-y1)^{2} }[/tex]
=[tex]\sqrt{(-1-(-2))^{2}+(3-1)^{2} }[/tex]
=[tex]\sqrt{1^{2}+2^{2} }[/tex]
=√5 or 2.24
what is 3125 divided by 25
Answer:
125
Step-by-step explanation:
By using normal division method,
3125÷25 goes as 125
What is the approximate circumference of a circle that has a diameter of 379? Use 3.14 for π and express your answer to the hundredths place.
fill in the blank
__
thank you
The circumference of a circle with diameter 379 can be found using the formula:
C = πd
where d is the diameter of the circle. Substituting the given value of d, we get:
C = 3.14 x 379
C ≈ 1191.06
Rounding to the hundredths place, the circumference is approximately 1191.06 units of length (e.g. inches, centimeters, etc.).
Therefore, the answer to fill in the blank is 1191.06.
from independent surveys of two populations, 90% confidence intervals for the population means are constructed. what is the probability that neither interval contains the respective population mean? that both do?
The probability that neither interval contains the respective population mean is 0.01 and the probability that both intervals contain their respective population means is 0.81.
Assuming that the two populations are independent, we can use the fact that the confidence interval is constructed such that there is a 90% probability that the true population mean falls within the interval.
Let's denote the two populations as Population A and Population B, and the confidence intervals for their respective means as CI_A and CI_B.
The probability that neither interval contains the respective population mean can be calculated as the complement of the probability that at least one interval contains its respective population mean.
P(neither interval contains its respective population mean) = 1 - P(at least one interval contains its respective population mean)
To calculate the probability of at least one interval containing its respective population mean, we can use the fact that the probability of an event A or B occurring is equal to the sum of their individual probabilities minus the probability of both events occurring simultaneously:
P(A or B) = P(A) + P(B) - P(A and B)
In this case, the event A is the event that CI_A contains the true population mean for Population A, and event B is the event that CI_B contains the true population mean for Population B.
Using this formula, we can calculate:
P(at least one interval contains its respective population mean) = P(CI_A contains true mean for Population A) + P(CI_B contains true mean for Population B) - P(both intervals contain their respective population means)
Since the two populations are independent, the events of each interval containing its respective population mean are also independent. Therefore, the probability of both intervals containing their respective population means can be calculated as the product of their individual probabilities:
P(both intervals contain their respective population means) = P(CI_A contains true mean for Population A) * P(CI_B contains true mean for Population B)
Now we have all the information we need to calculate the probability of interest:
P(neither interval contains its respective population mean) = 1 - P(at least one interval contains its respective population mean)
= 1 - [P(CI_A contains true mean for Population A) + P(CI_B contains true mean for Population B) - P(CI_A contains true mean for Population A) * P(CI_B contains true mean for Population B)]
Note that we don't have any specific information about the probabilities of the intervals containing their respective population means, so we can't calculate this probability exactly. However, we do know that the confidence intervals are constructed such that there is a 90% probability that the true population mean falls within the interval. This means that the probability of each interval containing its respective population mean is 0.9.
Using this information, we can calculate:
P(neither interval contains its respective population mean) = 1 - [0.9 + 0.9 - 0.9 * 0.9]
= 1 - 0.99
= 0.01
Therefore, the probability that neither interval contains the respective population mean is approximately 0.01, assuming that the confidence intervals were constructed using a 90% confidence level.
Similarly, the probability that both intervals contain their respective population means can be calculated as:
P(both intervals contain their respective population means) = P(CI_A contains true mean for Population A) * P(CI_B contains true mean for Population B)
= 0.9 * 0.9
= 0.81
Therefore, the probability that both intervals contain their respective population means is approximately 0.81, assuming that the confidence intervals were constructed using a 90% confidence level.
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(3.6*10^-5)/(1.8*10^2)
Answer:
2*10^-7
Step-by-step explanation:
3.6*10^-5 is the same thing as [tex]\frac{1}{3.6*10^5}[/tex]. So [tex]\frac{1}{3.6*10^5}*\frac{1}{1.8*10^2}[/tex] is equal to [tex]\frac{1}{2*10^7}[/tex]which is equal to [tex]2*10^-7[/tex]
The star basketball player at your high school claims to make 50% of his three-point shots. Recently, he missed 7 of his first 10 three-point shots. Is this a surprising result if the player’s claim is true? Assume that the player is, in fact, able to make 50% of his three-point shots.
We need to carry out a simulation using a die to estimate the probability that he would miss 7 or more of his first 10 three-point shots.
answers
✔ even = make; odd = miss
✔ 100 times
The star basketball player at your high school claims to make 50% of his three-point shots. Recently, he missed 7 of his first 10 three-point shots. Is this surprising for this player? Assume that the player is, in fact, able to make 50% of his three-point shots. We need to carry out a simulation to estimate the probability that he would miss 7 or more of his first 10 three-point shots.
Here are the results of 100 simulated trials. Which of the following answers the question of interest?
answer: Based on the 100 simulated trials, there is about an 18% chance of the player missing 7 of his first 10 three-point shots. This probability is not unusually low, so it is not unlikely.
The probability that the player would miss 7 or more shots is given as follows: 17%.
As the probability is greater than 5%, it would not be a surprise.
What is the binomial distribution formula?The binomial distribution formula gives the probability of obtaining a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success is constant throughout the trials.
The mass probability formula is given by the equation presented as follows:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters, along with their meaning, are listed as follows:
n is the fixed number of independent trials.p is the constant probability of a success on a single independent trial of the experiment.The parameter values for this problem are given as follows:
n = 10, p = 0.5.
Hence the probability of 7 or more misses is given as follows:
P(X >= 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10).
Using a calculator, with the given parameters, the probability is given as follows:
P(X >= 7) = 0.17 = 17%.
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If it is known that a and b are positive integers and that a-b=2 evaluate the following.
Answer: 1/9
Step-by-step explanation: I do rsm it is correct from my hw
A conic storage unit has a radus of 8 feet and a height equal to its diameter.
What is the volume of the storage unit?
Answer:
Step-by-step explanation:
he height of the storage unit is equal to twice its radius (since the diameter is twice the radius), so the height is 2 x 8 = 16 feet.
The storage unit is in the shape of a cylinder, so we can use the formula for the volume of a cylinder, V = πr^2h, where r is the radius and h is the height:
V = π(8^2)(16)
V = π(64)(16)
V = 3,218.69 cubic feet (rounded to two decimal places)
Therefore, the volume of the storage unit is approximately 3,218.69 cubic feet.
I need helpp ASAP….please
Answer:
the formula is y=0.06x + 45
A. The rate of change is 0.06 dollars per kilometer
B. The initial cost is $45 dollars
C. The charge after 837 km is 95.22 including the initial cost
D. For $200 dollars you could drive 2583.3 km repeating.
Step-by-step explanation:
Answer:
Step-by-step explanation:
Your initial equations are: r is rate i is initial fee
59.4 = r(240) + i
74.4 = r(490) + i
solve the systems of equations. I eliminate i by multiplying both sides of top equation by -
74.4 = r(490) + i
-59.4 = -r(240) - i
15=250r
r=.06
plug r back in to get i
59.4=(.06)(240)+i
i=45
so
y=.06x+45
amanda wants to create a scale drawing of a rectangular porch that is 5 feet by 12 feet. The scale drawing needs to be as large as possible and fit on a sheet of paper that is 6 inches by 6 inches
The dimensions of the porch and the size of the paper indicates;
The appropriate scale for the drawing is the option 1 inch to 2 feet
1 in. : 2 feet
What is a scale drawing?A scale drawing is a drawing which has been increased or reduced compared to the size of the original or initial drawing.
The dimensions of the rectangular porch = 5 feet by 12 feet
The size of the sheet of paper = 6 inches by 6 inches
The limiting size from the actual porch is the length of 12 feet
Therefore;
The ratio of the length of the corresponding size of the drawing to the size actual porch is therefore;
6 inches is equivalent to 12 feet, which indicates;
6/6 = 1 inch is equivalent to 12/6 = 2 feet
The ratio is therefore; 1 in. : 2 ftThe possible question, obtained from a similar question posted on the net is to select the appropriate scale of the drawing
The possible options are;
1 in. : 2 ft.
1 in. : 2.5 ft.
6 in. : 2.5 ft.
2.5 in. : 6 ft.
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Can someone help me please?
IF 4 dice are thrown, what is the probability of getting
i.exactly 3sixes
ii.exactly2sixes
iii.no sixes.
When rolling a fair six-sided die, the probability of rolling a six is 1/6. Assuming that the dice are fair and independent, we can use the rules of probability to find the probability of rolling certain outcomes when 4 dice are thrown:
i. To find the probability of getting exactly 3 sixes when 4 dice are thrown, we can use the binomial probability formula:
P(exactly k successes in n trials) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials, k is the number of successful outcomes, p is the probability of success on each trial, and (n choose k) is the binomial coefficient.
For this problem, n = 4, k = 3, and p = 1/6.
P(exactly 3 sixes) = (4 choose 3) * (1/6)^3 * (5/6)^1
= 4 * (1/216) * (5/6)
= 5/54
Therefore, the probability of getting exactly 3 sixes when 4 dice are thrown is 5/54.
ii. To find the probability of getting exactly 2 sixes when 4 dice are thrown, we can use the same formula with k = 2:
P(exactly 2 sixes) = (4 choose 2) * (1/6)^2 * (5/6)^2
= 6 * (1/36) * (25/36)
= 25/72
Therefore, the probability of getting exactly 2 sixes when 4 dice are thrown is 25/72.
iii. To find the probability of getting no sixes when 4 dice are thrown, we can use the complement rule:
P(no sixes) = 1 - P(at least one six)
To find the probability of getting at least one six, we can use the complement of getting no sixes:
P(at least one six) = 1 - P(no sixes)
For each die, the probability of not rolling a six is 5/6. Therefore, the probability of getting no sixes on 4 dice is:
P(no sixes) = (5/6)^4
= 625/1296
Therefore, the probability of getting no sixes when 4 dice are thrown is 625/1296, and the probability of getting at least one six is 1 - 625/1296 = 671/1296.
