Answer:
[tex]$(x-h)^2+(y-k)^2= (r+ 40 )^2$[/tex]
Step-by-step explanation:
It is given that a meteorologist is using a Doppler radar to find out the distance of the storm as it passes using a polar grid.
The origin of the radar screen is its center.
And the distance between each ring is 10 miles from the center.
So let :
r = radius
h and k = be the coordinates of the center.
So the equation of a circle is :
[tex]$(x-h)^2+(y-k)^2= r^2 $[/tex]
Now the equation of the fourth circle when the radius changes by 10 miles is :
[tex]$(x-h)^2+(y-k)^2= (r+ 40 )^2$[/tex]
The mean grade in Mathematics of Grade 11 students from school A is 83 with standard deviation of 4, while the mean grade in Mathematics of Grade 11 students from school B is 87 with standard deviation of 5. Using 49 samples from school A and 64 from school B, what is the probability that the mean grade of students from school B exceeds the mean grade of students from school A by at least 3 but less than 5?
Using the context surrounding the word trough in line 9 of "Concrete Mixers," explain what a trough looks like and what it does on a concrete mixer. .
sorry I can't delete it this not answer your question
Out of 28 students, 12 have at least one brother and 13 have at least one sister. 8 students have both brothers and sisters. How many students do not have either a brother or a sister?
Answer:
The answer is 11
Step-by-step explanation:
brother only=12-8=4
sister only=13-8=5
brother and sister =8
neither of them=x
28=4+8+5+x
28=17+x
x=28-17
x=11
Answer:
11
Step-by-step explanation:
You want to know the number of students with no brother or sister, given that 12 of 28 have at least one brother, 13 have at least one sister, and 8 have both.
Two-way tableThe attachment shows the two-way table that can be formed from the given information. The four given numbers are shown in green at lower right. The other numbers in the table are filled in to make the totals accurate.
The number of students not having either a brother or sister is 11.
__
Additional comment
In terms of probability, which is the fraction that each event is of the total, you have ...
P(B∪S) = P(B) +P(S) -P(B∩S) . . . . . . . useful relation to memorize
P(B'∩S') = 1 -P(B∪S) = 1 -(12/28 +13/28 -8/28) = 11/28
This tells you 11 of the 28 students do not have a brother or sister.
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Can I have some help I don't really understand this question.
Answer:
40 square units
Step-by-step explanation:
This is a rhombus, so you multiply the two diagonals and divide by 2.
8x10÷2=40
Can someone please give me this answer
Answer:
42°
Step-by-step explanation:
According to Exterior Angle Theorem,
an exterior angle of a ∆ is equal to the sum of the opposite interior angles.
So,
h+96°=138°
h=138°-96°
h=42°
Find the value of each variable
The value of angle k is determined as 60⁰.
The value of angle m is determined as 120⁰.
What is the measure of angle m and angle k?The measure of angle m and k is calculated by applying the following theorem as follows;
The value of angle k is calculated as;
60 + 2k = 180 (opposite angles of a cyclic quadrilateral are supplementary)
60 + 2k = 180
2k = 180 - 60
2k = 120
k = 120/2
k = 60⁰
The value of angle m is calculated as;
m + 60 = 180 (opposite angles of a cyclic quadrilateral are supplementary)
m = 180 - 60
m = 120⁰
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please help me this is kinda hard
Answer: 14.4 ft^2
Step-by-step explanation: The formula for the area of a triangle is A=1/2(b)(h). B is the base of the triangle and h is the height. First solve for the top part of the triangle. A=1/2(2)(3.2). A=3.2 ft^2. The 2 triangles on top are congruent, so you can multiply the area for the 1st triangle by 2 to get the combined area of the 2 top triangles. 3.2*2=6.4 ft^2.
Now we will solve for the 2 bottom triangles. A=1/2(2)(4). A=4 ft^2. The 2 triangles on the bottom are also congruent, so multiply the area that we got by 2 to get the combined area of both bottom triangles. 4*2=8 ft^2.
Finally, add both values to get the total area. 6.4+8=14.4 ft^2.
Please help question in the picture
Answer:
64 m²
Step-by-step explanation:
Area of the trapezoid = ½(a + b)h
Where,
a = 6 m
b = 10 m
h = 8 m
Plug in the values
Area = ½(6 + 10)*8
Area = ½(16)*8
Area = 8*8
Area = 64 m²
Find the measurement of RS
134 degrees is equivalent to the measure of arc RS
Circle GeometryIn order to determine the measure of the arc RS, we will use the theorem below:
The measure of the angle at the vertex is half that of its intercepted arc. Based on the theorem, we can set up the equation:
arcRS = 2m<Q
arcRS = 2(67)
arcRS = 134 degrees
Hence the measure of the arc RS from the given circle is equivalent to 134 degrees
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An cylindrical oatmeal container has a diameter of 5 inches and a volume of 239.425
in. What measurement is closest to the lateral surface area of the oatmeal
container?
230.81 in 2
o
191.54 in.2
15.24 in.2
478.85 in.2
Answer:478.85 in2
Step-by-step explanation:
Just divide LOL
Michael is buying a pair of jeans that regularly cost $40. They are on sale for 15% off. If the tax rate is 8.5%, what is the finale price of the jeans? Write ONE equation and solve.
Answer:
139284 238
Step-by-step explanation:
Which number line shows the solution of - 5x + 20 < 35?
Answer:
15, 14, 13, 12, 11 and so on
Step-by-step explanation:
PLS HELP!! I can't figure it out
The dependent and independent variables are distance from destination and time respectively.
The relationship is linear as the change is constantrate of change is -2.05 mi/min139 minutes .The rate of changeRate of change = change in y/Change in x
Rate of change= (244-285)/(20-0)
Rate = -2.05
Helicopter's DestinationWhen the helicopter reaches its destination , y = 0
We can write the traveling equation in the form y = mx + c
Where :
c= intercept ; m = slope
y = -2.05x + 285
At y = 0
0 = -2.05x + 285
-2.05x = - 285
x = 285/2.05
x = 139.02
Hence, the helicopter will reach its destination after 139 minutes .
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In 2006, a sample of 200 in-store shoppers showed that 42 paid by debit card. In 2009, a sample of the same size showed that 80 paid by debit card. (a) Formulate appropriate hypotheses to test whether the percentage of debit card shoppers increased. (b) Carry out the test at alpha
The percentage of debit shoppers has increased.
21% (2006) < 40% (2009)
In 2006, the percentage of debit shoppers is 21%.
In 2009, the percentage of debit shoppers is 40%.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
We have,
In 2006:
A sample of 200 in-store shoppers showed that 42 paid by debit card.
The percentage of debit shoppers in 2006.
= 42/200 x 100
= 42/2
= 21%
In 2009:
A sample of the same size showed that 80 were paid by debit card.
The percentage of debit shoppers in 2009.
= 80/200 x 100
= 40%
Thus,
The percentage of debit shoppers has increased.
21% (2006) < 40% (2009)
In 2006, the percentage of debit shoppers is 21%.
In 2009, the percentage of debit shoppers is 40%.
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What is the average rate of change of the function =2sin(1/2)on the interval [0, π]?
Answer:
4/x = [tex]\frac{\pi }{6}[/tex], 5 [tex]\frac{\pi }{6}[/tex]
Step-by-step explanation:
Sorry if I am wrong
The average rate of change of the function in the given interval [0, π] is [tex]\dfrac{2}{\pi }[/tex].
What is the average rate of change?The average Rate of Change of the function f(x) can be calculated as;
[tex]f(x) = \dfrac{f(b) - f(a)}{b-a}[/tex]
The given function is f(x) = 2sin(1/2)x on the interval [0, π]
Here a = 0
b = π
f(a) = 2sin(1/2)a
f(0) = 0
f(b) = 2sin(1/2)π = 2
Step 2: Find Average
[tex]f(x) = \dfrac{f(b) - f(a)}{b-a}\\\\f(x) = \dfrac{2- 0}{\pi }\\\\f(x) = \dfrac{2}{\pi }\\[/tex]
Therefore, the average rate of change of the function in the given interval [0, π] is [tex]\dfrac{2}{\pi }[/tex].
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Wendy bought a silver necklace online. It cost $23.15 plus 20% shipping and handling. What was the total cost?
The Wendy spent $27.78 on a silver necklace online.To calculate the total cost of the silver necklace, we need to add the cost of the necklace to the shipping and handling charges.
The cost of the necklace is given as $23.15.
Wendy bought a silver necklace online. It cost $23.15 plus 20% shipping and handling.
What was the total cost?When purchasing a silver necklace online, Wendy spent $23.15, but she had to pay for shipping and handling as well, which cost 20% of the total price of the item.
To find the total cost of the necklace, follow these steps:
1. Determine the cost of shipping and handling: $23.15 × 0.20 = $4.632
Add the cost of shipping and handling to the cost of the necklace:$23.15 + $4.63 = $27.78
The total cost of the necklace is $27.78.
It's important to consider the additional charges such as shipping and handling when calculating the total cost of an online purchase to have an accurate understanding of the amount spent.
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Two students in different classes took the same math test. Both students received a
score of 87. In student A's class the mean was 78 and the standard deviation of 5. In
student B's class the mean was 76 with a standard deviation of 4. Which student
scored in the top 10% of their class?
Answer:
Both students scored in the top 10% of their classes.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
In this question:
Top 10% = Above the 100 - 10 = 90th percentile.
The 90th percentile of scores is X when Z has a pvalue of 0.9, that is, Z = 1.28.
So, the student who had a z-score above 1.28 scored in the 90th percentile of their class.
In student A's class the mean was 78 and the standard deviation of 5. He scored 87.
We have that [tex]\mu = 78, \sigma = 5, X = 87[/tex]
Then
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{87 - 78}{5}[/tex]
[tex]Z = 1.8[/tex]
1.8 > 1.28, so student A scored in the top 10% of his/her class.
Student B's class the mean was 76 with a standard deviation of 4. Scored 87.
We have that [tex]\mu = 76, \sigma = 4, X = 87[/tex]
Then
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{87 - 76}{4}[/tex]
[tex]Z = 2.75[/tex]
2.75 > 1.28, so student B also scored in the top 10% of his/her class.
Both students scored in the top 10% of their classes.
3. The third side of a triangle measures (3x-5) cm. If the length of the midline is 14 m,
what is x?
A 8
C. 10
B. 9
D. 11
Answer:
D. 11
Step-by-step explanation:
Midsegment theorem:
The length of the midsegment of a triangle is half the length of it's third side.
In this question:
Third side: 3x - 5
Midsegment: 14m
So
[tex]\frac{3x - 5}{2} = 14[/tex]
[tex]3x - 5 = 28[/tex]
[tex]3x = 33[/tex]
[tex]x = \frac{33}{3}[/tex]
[tex]x = 11[/tex]
The correct answer is given by option D.
are these correct???
Answer:
They are all correct.
Step-by-step explanation:
1.
1/2(b)(h)
1/2(4)(6)
1/2(24)
12
2. l x w
7 x 5
35
3. (b)(h)
(8)(4)
32
PLEASE HELP ME
Drag the tiles to the boxes to form correct pairs. Not all tiles will be used.
Match the expressions with their simplified forms.
(a) √2 · √8. The simplified surd expression is 4.
(b) √80. The simplified surd expression is 4√5.
(c) √5/√20. The simplified surd expression is 1/2.
(d) √20. The simplified surd expression is 2√5.
What is the simplification of the surd expression?The given surd expression is simplified as follows;
(a) √2 · √8
we can simplify it as;
√2 · √8 = √(2 x 8) = √16 = 4
(b) √80
we can simplify it as;
√80 = √ (16 x 5) = √16 x √5 = 4√5
(c) √5/√20
we can simplify it as;
√5/√20 x √20 / √20
= (√5 x √20 ) /(20)
= √100 / 20
= 10/20
= 1/2
(d) √20
we can simplify it as;
√20 = √4 x √5 = 2√5
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A solid figure is composed of a cube and a right triangular
prism. The figure and some of its dimensions are shown in
this diagram.
- 8 cm
What is the volume of the figure?
A
6 cm
B
560 cubic centimeters
704 cubic centimeters
C 728 cubic centimeters
Answer:
Option B
Step-by-step explanation:
704 cubic centimeters
Determine whether 548 is greater than or less than 373. Then write the expression showing this using < or >.
Answer:
548 > 373
Step-by-step explanation:
548 is greater than 373 because when we compare the digits from left to right, we find that the first digit of 548 (5) is greater than the first digit of 373 (3). Therefore, we can conclude that 548 is greater than 373.
The ">" symbol is used to represent "greater than" in mathematical comparisons.
Hope this helps!
Printer A prints 100 pages for $23.99. Printer B prints 275 sheets for $63.99. Which printer has the better rate of cost per page?
Printer A, because the approximate rate of Printer A, $0.24 per page, is greater than the approximate rate of Printer B, $0.23 per page
Printer A, because the approximate rate of Printer A, $4.16 per page, is less than the approximate rate of Printer B, $4.30 per page
Printer B, because the approximate rate of Printer A, $0.24 per page, is greater than the approximate rate of Printer B, $0.23 per page
Printer B, because the approximate rate of Printer A, $4.16 per page, is less than the approximate rate of Printer B, $4.30 per page
PLEASE HURRY I NEEED THIS RN :)
Printer A has the better rate of cost per page because the approximate rate of Printer A, $0.24 per page, is greater than the approximate rate of Printer B, $0.23 per page.
How to calculate the rate?Rate demonstrates how many times one number can fit into another number. It contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).
In this case, Printer A prints 100 pages for $23.99. The rate is:
[tex]\sf = \dfrac{\$23.99}{100}[/tex]
[tex]\sf = 0.2399\thickapprox\$0.24 \ per \ page[/tex]
Printer B prints 275 sheets for $63.99. The rate is:
[tex]\sf = \dfrac{\$63.99}{275}[/tex]
[tex]\sf = 0.2327\thickapprox\$0.23 \ per \ page[/tex]
Therefore, based on the above calculations, we can see that Printer A has the better rate of cost per page because the approximate rate of Printer A, $0.24 per page, is greater than the approximate rate of Printer B, $0.23 per page.
So option (A) is correct.
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In the image below, line q is parallel to liner, and line s is a transversal.
D
9
B
What is the relationship between the measures of ZAand ZB?
The measures of Aand ZB are
becuase they are
Answer:
equal
alternate exterior angles
Step-by-step explanation:
The measure of given angles are equal to each other because they are alternate exterior angles,
Label the spinner below with the letters A though J. What is the probability of landing on a vowel on the spinner and rolling a number greater that 4 on the fair number cube?
Answer:
3 for vowles and two for number
Step-by-step explanation:
I also need help on this one just need points
HELP ASAP 6TH GRADE WORK LOOK AT PHOTO
The simplified exponential function is determined as x⁴.
What is the simplification of the exponential function?An exponential function is a mathematical function used to calculate the exponential growth or decay of a given set of data.
To simplify an exponential function, we will apply the rules of exponent as shown below;
The given exponential function;
= (∛ x² )⁶
The given expression is simplified as follows;
= [tex](x^2) ^{\frac{1}{3} \times 6}[/tex]
= ( x² )²
= x⁴
Thus, the simplified exponential function is determined by applying the rules of multiplication of powers.
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Write a quadratic equation with the given roots. Write the equation in the form a * x ^ 2 + bx + c = 0 where a, b, and c are
integers.
5/4 : and 9
The quadratic equation from the given root is 4x² - 41x + 45 = 0
Writing a quadratic equation from the given rootFrom the question, we have the following parameters that can be used in our computation:
Roots = 5/x and 4
The equation of the function can be calculated as
(x - root) * (x - root) = 0
using the above as a guide, we have the following:
(x - 5/4) * (x - 9) = 0
When expanded, we have
(4x - 5)(x - 9)/4 = 0
This gives
4x² - 5x - 36x + 45 = 4 * 0
So, we have
4x² - 41x + 45 = 0
Hence, the quadratic equation from the given root is 4x² - 41x + 45 = 0
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Find the volume of a cylinder that has a radius of 1/2and a height of 1.
Answer:
0.79
Step-by-step explanation:
Equilateriall triangle. Find the length of side X in simple radical form with a rational denominator
Answer:
x = 4
Step-by-step explanation:
since the triangle is equilateral then the vertex angles are congruent, each 60°
using the sine ratio in the right triangle with x as its hypotenuse and the exact value
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{\sqrt{12} }{x}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
x × [tex]\sqrt{3}[/tex] = 2[tex]\sqrt{12}[/tex] ( divide both sides by [tex]\sqrt{3}[/tex] )
x = [tex]\frac{2\sqrt{12} }{\sqrt{3} }[/tex] = 2 × [tex]\sqrt{\frac{12}{3} }[/tex] = 2 × [tex]\sqrt{4}[/tex] = 2 × 2 = 4
A small publishing company is planning to publish a new book. The production costs will include one-time fixed costs (such as editing) and variable costs (such as printing). There are two production methods it could use. With one method, the one-time fixed costs will total 44,907, and the variable costs will be per book. With the other method, the one-time fixed costs will total 22,907, and the variable costs will be 23.25 per book. For how many books produced will the costs from the two methods be the same?
Substituting the given variable cost per book for Method 1 (which is not specified), we can calculate the value of 'x' that makes the costs equal for the two methods.
Let's assume the number of books produced is denoted by 'x'. We need to find the value of 'x' for which the costs from the two methods are the same.
For the first method, the total cost is the sum of the one-time fixed cost and the variable cost per book:
Total cost for Method 1 = 44,907 + (variable cost per book) * x
For the second method, the total cost is the sum of the one-time fixed cost and the variable cost per book:
Total cost for Method 2 = 22,907 + (23.25 * x)
To find the number of books produced when the costs from the two methods are equal, we set the total costs equal to each other and solve for 'x':
44,907 + (variable cost per book) * x = 22,907 + (23.25 * x)
Subtracting (23.25 * x) from both sides and rearranging the equation:
21,000 = 23.25 * x - (variable cost per book) * x
21,000 = x * (23.25 - (variable cost per book))
Dividing both sides by (23.25 - (variable cost per book)):
x = 21,000 / (23.25 - (variable cost per book))
Substituting the given variable cost per book for Method 1 (which is not specified), we can calculate the value of 'x' that makes the costs equal for the two methods.
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NO LINKS!! URGENT HELP PLEASE!!
Find the probability of each event.
28. One day, 9 babies are born at a hospital. Assuming each baby has an equal chance of being a boy or girl, what is the probability that exactly 4 of the 9 babies are girls?
29. A gambler places a bet on a horse race. To win, he must pick the top 3 finishers in order. 13 horses of equal ability are entered in the race. Assuming the horses finish in random order, what is the probability that the gambler will win his bet?
Answer:
28. 0.2461 or 24.61%.
29. 0.00058275 or 0.058275%.
Step-by-step explanation:
Question 28:
we can use the binomial probability formula to calculate the probability:
[tex]\boxed{\bold{P(X = k) = (nCk) * p^k * (1 - p)^(n - k)}}[/tex]
Where:
P(X = k) is the probability of getting exactly k successesnCk is the number of combinations of n items taken k at a timep is the probability of a single successn is the total number of trialsIn this case, we have n = 9 babies, and each baby has a 50% chance of being a girl (p = 0.5). We want to find the probability that exactly 4 of them are girls (k = 4).
Using the formula, we can calculate the probability as follows:
[tex]\bold{P(X = 4) = (9C4) * (0.5)^4 * (1 - 0.5)^(9 - 4)}[/tex]
Calculating the values:
(9C4) = 126 (0.5)^4 = 0.0625(1 - 0.5)^(9 - 4) = 0.5^5 = 0.03125Now, we can substitute these values into the formula:
P(X = 4) = 126 * 0.0625 * 0.03125 = 0.2461 or 24.61%
Therefore, the probability that exactly 4 out of 9 babies are girls is approximately 0.2461 or 24.61%.
Question 29:
We need to calculate the number of possible outcomes to calculate this probability, where the gambler correctly predicts the top 3 finishers in order and divides it by the total number of possible outcomes.
The total number of possible outcomes is the number of permutations of 13 horses taken 3 at a time.
This can be calculated as:
[tex]13P3 = \frac{13! }{(13 - 3)! }= \frac{13! }{ 10! }=\frac{13*12*11*10! }{ 10! }= 13 * 12 * 11 = 1,716[/tex]
Now,
To calculate the number of favorable outcomes where the gambler predicts the top 3 finishers correctly, we need to consider that there is only one correct order for the horses to finish.
Therefore, there is only one favorable outcome.
The probability of the gambler winning his bet is given by:
[tex]\boxed{\bold{P\:(winning) = \frac{Number\: of \:favorable\: outcomes }{ Total \:number \:of \:outcomes}}}[/tex]
[tex]P(winning) = \frac{1 }{1,716}=0,00058275 \: or\:0.058275%[/tex]
Therefore, the probability that the gambler will win his bet is approximately 0.00058275 or 0.058275%.
Answer:
28) 0.246 = 24.6%
29) 1/286 = 0.350%
Step-by-step explanation:
Question 28We can model the given scenario as a binomial distribution.
Binomial distribution[tex]X \sim \text{B}(n,p)[/tex]
where:
X is the random variable that represents the number of successes.n is the fixed number of independent trials.p is the probability of success in each trial.Given the probability that a baby is born a girl is 0.5, and the number of babies is 9:
[tex]\boxed{X \sim \text{B}(9,0.5)}[/tex]
where the random variable X represents the number of babies who are girls.
To find the probability that at exactly 4 babies are girls, we need to find P(X = 4).
To do this, we can use the binomial distribution formula:
[tex]\boxed{\displaystyle \text{P}(X=x)=\binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}}[/tex]
Substitute the values of n = 9, p = 0.5 and x = 4 into the formula:
[tex]\begin{aligned}\displaystyle \text{P}(X=4)&=\binom{9}{4} \cdot 0.5^4 \cdot (1-0.5)^{9-4}\\\\&=\dfrac{9!}{4!\:(9-4)!} \cdot 0.5^4 \cdot 0.5^{5}\\\\&=126 \cdot 0.0625 \cdot 0.03125\\\\& = 0.24609375\end{aligned}[/tex]
Therefore, the probability that 4 babies from a sample of 9 babies are girls is 0.246 (3 s.f.) or 24.6%.
We can also use the binomial probability density function of a calculator to calculate P(X = 4).
Inputting the values of n = 9, p = 0.5 and x = 4 into the binomial pdf:
[tex]\text{P}(X=4)=0.24609375[/tex]
Therefore, this confirms that the probability that 4 babies from a sample of 9 babies are girls is 0.246 (3 s.f.) or 24.6%.
[tex]\hrulefill[/tex]
Question 29To calculate the probability that the gambler will win his bet, we need to determine the number of favorable outcomes (winning combinations) and the total number of possible outcomes.
The gambler wins if he picks the top three horses in any order. There are 6 ways for the three winners to be arranged in the top three.
There are a total of 13 horses in the race.
The number of ways to choose the first-place horse is 13. After the first-place horse is chosen, there are 12 remaining horses, so the number of ways to choose the second-place horse is 12. Finally, after the first two horses are chosen, there are 11 remaining horses, so the number of ways to choose the third-place horse is 11.Therefore, the total number of possible outcomes is:
[tex]13 \times 12 \times 11 = 1716[/tex]
Therefore, the probability that the gambler will win his bet is:
[tex]\begin{aligned} \sf Probability &=\sf \dfrac{Favorable \;outcomes}{Total\;outcomes}\\\\&=\dfrac{6}{1716}\\\\&=\dfrac{1}{286}\\\\ & \approx0.350\%\; \sf (3\;d.p.)\end{aligned}[/tex]