The probability that at least two policies will pay out before the insured parties have reached age 61 is 0.3400.
The answer is as follows: The expected number of policies that will pay out before the insured parties have reached age 61 isE(x) = np = 12 × 0.05 = 0.6Therefore, the expected number of policies that will pay out before the insured parties have reached age 61 is 0.6.b) The variance of the number of policies that will pay out before the insured parties have reached age 61 isVar(x) = np(1-p) = 12 × 0.05(1-0.05) = 0.570Therefore, the variance of the number of policies that will pay out before the insured parties have reached age 61 is 0.570.c) To find out the probability that at least two policies will pay out before the insured parties have reached age 61, we can use the complement rule that is:P(X ≥ 2) = 1 - P(X < 2) = 1 - P(X = 0) - P(X = 1)P(X = 0) = (0.95)^12 = 0.2835 (probability that none of the 12 policies will pay out before the insured parties have reached age 61)P(X = 1) = 12C1(0.05)(0.95)^11 = 0.3765 (probability that only one of the 12 policies will pay out before the insured parties have reached age 61)Therefore,P(X ≥ 2) = 1 - P(X = 0) - P(X = 1) = 1 - 0.2835 - 0.3765 = 0.3400Therefore, the probability that at least two policies will pay out before the insured parties have reached age 61 is 0.3400.
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5. Solve the quadratic equation 14(x - 1)2-(x-1)-3=0
Answer: x=10/9
Step-by-step explanation:
hw06-MoreProbability: Problem 10 (1 point) Three dice are tossed. Find the probability of rolling a sum greater than 5 . Answer: You have attempted this problem 0 times. You have unlimited attempts remaining.
There are 6 + 9 + 9 + 12 + 9 + 9 + 6 = 60 outcomes where the sum of the three dice is greater than 5.
So, the probability of rolling a sum greater than 5 is given by:
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 60 / 216
Probability = 5 / 18
We know that when a dice is rolled, the numbers that come up on the dice are 1, 2, 3, 4, 5 and 6. Since there are three dice, the total number of possible outcomes when they are tossed is given by 6 * 6 * 6 = 216.
Now we have to find the probability of rolling a sum greater than 5. To find this probability, we need to consider all the cases where the sum of the three dice is greater than 5.
The possible outcomes where the sum of the three dice is greater than 5 are:
Sum of 6: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) (Total 6)
Sum of 7: (1, 2, 4), (1, 4, 2), (2, 1, 4), (2, 4, 1), (4, 1, 2), (4, 2, 1), (1, 3, 3), (3, 1, 3), (3, 3, 1) (Total 9)
Sum of 8: (1, 2, 5), (1, 5, 2), (2, 1, 5), (2, 5, 1), (5, 1, 2), (5, 2, 1), (3, 3, 2), (3, 2, 3), (2, 3, 3) (Total 9)
Sum of 9: (1, 3, 5), (1, 5, 3), (3, 1, 5), (3, 5, 1), (5, 1, 3), (5, 3, 1), (4, 2, 3), (4, 3, 2), (2, 4, 3), (3, 4, 2), (2, 3, 4), (3, 2, 4) (Total 12)
Sum of 10: (1, 4, 5), (1, 5, 4), (4, 1, 5), (4, 5, 1), (5, 1, 4), (5, 4, 1), (2, 4, 4), (4, 2, 4), (4, 4, 2) (Total 9)
Sum of 11: (1, 5, 5), (5, 1, 5), (5, 5, 1), (2, 5, 4), (2, 4, 5), (4, 5, 2), (4, 2, 5), (5, 4, 2), (5, 2, 4) (Total 9)
Sum of 12: (3, 4, 5), (3, 5, 4), (4, 3, 5), (4, 5, 3), (5, 3, 4), (5, 4, 3) (Total 6)
Therefore, there are 6 + 9 + 9 + 12 + 9 + 9 + 6 = 60 outcomes where the sum of the three dice is greater than 5.
So, the probability of rolling a sum greater than 5 is given by:
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 60 / 216
Probability = 5 / 18
Hence, the probability of rolling a sum greater than 5 is 5 / 18.
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Angles and Parallel Lines Two parallel lines are cut by a transversal as shown in the image. Question 1 Find the measure of angle A. Responses A 150°150° B 115°115° C 125°125° D 130°
Tim saw that a pair of $200 dollar jeans had been marked down by 20%. he told his friend that if you took the new price of the jeans and increased that value by 20%, the jeans would return to the original price of $200. do you agree with tim? show your calculations.
Answer: Let's use algebra to check if Tim is correct.
Let x be the original price of the jeans.
When the jeans are marked down by 20%, the new price becomes:
x - 0.2x = 0.8x
Tim claims that if we increase the new price by 20%, we will get the original price:
0.8x + 0.2(0.8x) = x
Simplifying the left-hand side:
0.8x + 0.16x = x
0.96x = x
This is not true for all values of x, which means that Tim's claim is not correct.
Let's use the given information that the original price is $200 to find out what the new price is after the 20% discount:
New price = Original price - Discount
New price = $200 - 20% of $200
New price = $200 - $40
New price = $160
If we increase the new price of $160 by 20%, we get:
New price + 20% of new price
$160 + 20% of $160
$160 + $32
$192
This is not equal to the original price of $200, so we can conclude that Tim's claim is incorrect.
Construct ΔPOR, Where PQ = 7.5cm, QR = 5cm and ∠Q = 90º
Using the given information, we found the angles of the right triangle as:
∠R = 56.30º
∠P = 33.7º
∠Q = 90º
What is a right triangle?
A triangle with a right angle, also known as a right-angled triangle or right-angled triangle, more technically an orthogonal triangle, has two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle. A right triangle's hypotenuse is its longest side, its "opposite" side is the one that faces a certain angle, and its "adjacent" side is the one that faces the angle in question. To describe the sides of right triangles, we utilise specific terminology. The side opposite the right angle is always the hypotenuse of a right triangle.
Given one of the angles of the triangles as 90º.
So it is a right triangle.
PQ = 7.5cm
QR = 5cm
∠Q = 90º
The figure is given below.
Now PR is the hypotenuse.
Hypotenuse = [tex]\sqrt{base^2+height^2}[/tex]
base = 5 = QR
height = 7.5 = PQ
PR = [tex]\sqrt{5^2+7.5^2} = \sqrt{81.25}[/tex] = 9.01 cm
We can find the angles using trigonometric relations.
sin R = PQ/PR = 7.5/9.01 = 0.832
∠R = 56.30º
∠P = 180 - (∠Q+∠R) = 180 - (90+56.30) = 33.7º
Therefore using the given information, we found the angles of the right triangle as:
∠R = 56.30º
∠P = 33.7º
∠Q = 90º
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The graph below belongs to which function family?
linear
quadratic
cubic
absolute value
It is desired to estimate the daily demand (sale) of a product registered by a company. For this, 12 days are selected at random with the following values in thousands for the demand
35, 44, 38, 55, 33, 56, 60, 45, 48, 40, 45, 35,42
Determine the population, the variable of interest and obtain the confidence interval for the average daily demand at a confidence level of 97%.
Answer:
Population: The population is the total demand (sale) of the product over all days.
Variable of interest: The variable of interest is the daily demand (sale) of the product.
To obtain the confidence interval for the average daily demand at a confidence level of 97%, we can use the following formula:
Confidence interval = sample mean ± (t-value x standard error)
where t-value is the value from the t-distribution for the desired confidence level and degrees of freedom, and the standard error is calculated as:
standard error = sample standard deviation / √n
where n is the sample size.
Using the given data, we can calculate:
Sample mean = (35+44+38+55+33+56+60+45+48+40+45+35+42)/12 = 44.5
Sample standard deviation = 9.92
Degrees of freedom = n-1 = 12-1 = 11
From the t-distribution table with 11 degrees of freedom and a confidence level of 97%, the t-value is approximately 2.718.
Therefore, the confidence interval for the average daily demand is:
Confidence interval = 44.5 ± (2.718 x 9.92/√12) = 44.5 ± 9.14
The lower limit is 44.5 - 9.14 = 35.36 and the upper limit is 44.5 + 9.14 = 53.64.
So, we can say with 97% confidence that the true population average daily demand falls within the range of 35.36 to 53.64 thousand units.
How do I find the lengths of sides that are cut by an altitude? (right triangle, the sides that the arrows are pointing at)
The length of line JC is 20 miles.
What is Pythagorean theorem?
The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of sides of triangle.
To find the length of line JC, which is the hypotenuse of the right triangle JSC, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides (the legs) is equal to the square of the length of the longest side (the hypotenuse).
In this case, we have:
JS² + SC² = JC²
Substituting the given values, we get:
12² + 16² = JC²
144 + 256 = JC²
400 = JC²
Taking square root both sides, we get:
JC = √400JC = 20
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If QV= 14 then what is the length of QU? and If QV= 14 then what is the length of QU? and If RV = 17 then what is the length of VS?
Answer:
QU is 21, and I *think* that VS would be 8.5.
Answer:
QV = 21 , VS = 8.5
Step-by-step explanation:
QU and RS are medians of Δ PQR
the point V where the medians intersect is the centroid.
on each median the distance from the vertex to the centroid is twice as long as the distance from the centroid to the midpoint.
then
VU = [tex]\frac{1}{2}[/tex] QV = [tex]\frac{1}{2}[/tex] × 14 = 7
so
QU = QV + VU = 14 + 7 = 21
and
VS = [tex]\frac{1}{2}[/tex] RV = [tex]\frac{1}{2}[/tex] × 17 = 8.5
Look at the following table then answer the questions below
a. Which of the functions in the table appears to be exponential?
b. What reasoning would you use to justify your answer?
c. Which function(s) would most likely model bacterial growth in a lab culture? Justify your reasoning.
d. Which values would most likely model a tub collecting water from a leaky faucet? Justify your reasoning.
For instance, f(x) is equal to 0.5 when x = 1 and equal to 1 when x = 2, function indicating that the amount of water collected rises by 0.5 for each unit increase in time.
what is function?Mathematicians research numbers, their variants, equations, forms, and related structures, as well as possible locations for these things. The relationship between a group of inputs, each of which has a corresponding output, is referred to as a function. Every input contributes to a single, distinct output in a connection between inputs and outputs known as a function. A domain, codomain, or scope is assigned to each function. Often, functions are denoted with the letter f. (x). The key is an x. There are four main categories of accessible functions: on functions, one-to-one capabilities, so many capabilities, in capabilities, and on functions.
A. It appears that the function f(x) = 2x is exponential.
b. The function may be expressed as f(x) = a * bx, where a denotes the starting value, b the growth factor, and x the input value. As can be seen from the table, the values of f(x) = 2x are exponentially growing by a factor of 2 for each input.
d. A linear function may be used to simulate a tub that collects water from a leaking faucet since the amount of water collected grows steadily over time.
For instance, f(x) is equal to 0.5 when x = 1 and equal to 1 when x = 2, indicating that the amount of water collected rises by 0.5 for each unit increase in time.
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The population of Wills Town decrease 8% over a 20-year. The population is currently 320,000 thousand what was the population of the Town 20 years ago.
What is the most specific category this shape can be sorted into?
A Venn diagram titled Triangles. Inside the diagram are circles, there is one labeled Scalene and one labeled Isosceles. Inside the Isosceles circle is another circle labeled Equilateral. Below the diagram is a triangle with a single tick mark on each side.
Equilateral
Isosceles
Scalene
Triangle
The most specific category that this shape can be sorted into is "Triangle."
What is the triangle?
A triangle is a three-sided polygon, which has three vertices. The three sides are connected with each other end to end at a point, which forms the angles of the triangle. The sum of all three angles of the triangle is equal to 180 degrees.
The Venn diagram shows the relationships between different types of triangles.
The Scalene circle represents all the triangles that have no equal sides.
The Isosceles circle represents all the triangles that have two equal sides.
The Equilateral circle represents all the triangles that have three equal sides.
The triangle with a tick mark on each side does not provide enough information to determine whether it is a scalene, isosceles, or equilateral triangle.
Therefore, the most specific category that this shape can be sorted into is "Triangle."
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Chocolate bar A weighs 80 grams and costs $1.00. Chocolate bar B weighs 85 grams and costs $1.20. Which is the best value and why?
Answer:
To determine the best value between chocolate bar A and B, we need to calculate the cost per gram of each chocolate bar.
For chocolate bar A, the cost per gram is:
$1.00 ÷ 80 grams = $0.0125 per gram
For chocolate bar B, the cost per gram is:
$1.20 ÷ 85 grams = $0.0141 per gram
Therefore, chocolate bar A is the better value as it costs less per gram compared to chocolate bar B. While chocolate bar B may weigh slightly more, its higher cost per gram means that you are paying more for each gram of chocolate compared to chocolate bar A.
When Casey woke up to get ready to go to school, he saw that the temperature was negative five
degrees. Casey knew when he went to bed it was twelve degrees warmer, What was the temperature
when Casey went to bed?
Number Sentence:
Answer:
Based on the difference between the temperature when Casey went to bed and when he woke up, the temperature when he went to bed was 7 degrees, which was 12 degrees warmer than -5 degrees.
What is the difference in temperature?The temperature difference is determined using subtraction.
Subtraction is one of the four basic mathematical operations, involving the minuend, the subtrahend, and the result of the operation called the difference.
The temperature when Casey woke up to prepare for school = -5
The difference between the temperature when Casey went to bed and when he woke up = 12 degrees warmer.
The temperature when Casey went to bed = 7 degrees (12 - 5)
Thus, we can conclude that Casey had a temperature of 7 degrees when he went to bed but woke up when it was -5 degrees.
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Find the distance between the two points in simplest radical form.
(−3,6) and (−8,−6)
Answer:
13
Step-by-step explanation:
To find the distance between two points in a coordinate plane, we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Using this formula, we can find the distance between the points (-3, 6) and (-8, -6) as follows:
d = sqrt((-8 - (-3))^2 + (-6 - 6)^2)
= sqrt((-5)^2 + (-12)^2)
= sqrt(25 + 144)
= sqrt(169)
= 13
Therefore, the distance between the two points in simplest radical form is 13.
Aaron is 8 years older than Judi. Judi is twice as old as Maree. All their ages add up to 43. What are their ages?
If Aaron is 8 years older than Judi, then Aaron's age is 22 years , Judi's age is 14 years and Maree's age is 7 years .
Let Maree's age be = M;
Judi is twice as old as Maree, which means ⇒ Judi's age is 2M;
And Aaron is 8 years older than Judi, which means
⇒ Aaron's age is (2M+8);
We know that the sum of their ages is 43, so we can write an equation:
⇒ M + 2M + (2M + 8) = 43
Simplifying and solving for M:
We get,
⇒ 5M + 8 = 43
⇒ 5M = 35
⇒ M = 7
So, Maree's age is 7 years.
Now, we use this to find Judi's and Aaron's ages:
⇒ Judi = 2M = 2 × 7 = 14
⇒ Aaron = 2M + 8 = 2 × 7 + 8 = 22
Therefore, Judi is 14 years old and Aaron is 22 years old.
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A rectangle mural measures 234 inches inches by 245. Rhiannon creates a new mural that is 33 inches longer
The new mural dimensions are 267 inches by 273 inches.To find the new dimensions, we must add 33 inches to the original length of 234 inches, giving us a new length of 267 inches.
To calculate the new dimensions of Rhiannon's mural, we must first identify the original dimensions of the mural which are 234 inches by 245 inches. To find the new dimensions, we must add 33 inches to the original length of 234 inches, giving us a new length of 267 inches. We must also add 28 inches to the original width of 245 inches, giving us a new width of 273 inches. Therefore, the new dimensions of Rhiannon's mural are 267 inches by 273 inches.
The complete question is :
A rectangle mural measures 234 inches by 245. Rhiannon creates a new mural that is 33 inches longer. What are the dimensions of Rhiannon's new mural?
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Y=4x/7
Write the ratio x:y in its simplest form
The ratio x:y in its simplest form is 7:4.
The ratio x:y can be found by dividing x by y. In this case, we can start with the equation y = 4x/7 and solve for x in terms of y.
Multiplying both sides by 7, we get:
7y = 4x
Dividing both sides by 4, we get:
x = 7y/4
Now we can substitute this expression for x into the ratio x:y:
x:y = (7y/4):y
Simplifying by canceling out the y, we get:
x:y = 7/4
So the ratio x:y in its simplest form is 7:4.
This means that for every 7 units of x, there are 4 units of y. The ratio cannot be simplified further because 7 and 4 do not have any common factors other than 1.
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Suppose you have a sample of 100 observations and you construct the empirical cumnlative distribution function (ECDF) based on this sample. What is the value of the ECDF at the smallest observation in the sample? 0 1/100 1/2 Not enough information to determine Empirical cumbotive Distribution fundion
ECDF=1/100
The value of the empirical cumulative distribution function (ECDF) at the smallest observation in a sample of 100 observations is 1/100.What is the empirical cumulative distribution function (ECDF)?The empirical cumulative distribution function (ECDF) is a graph that depicts the distribution of a given data set. It is used to illustrate the proportion or percentage of data points that fall below a particular value in the distribution. The ECDF can be used to construct a distribution function for data that do not have an existing theoretical distribution.The ECDF value at the smallest observation in a sample of 100 observations is 1/100. This is because the ECDF takes on a value of 0 for all values that are smaller than the smallest observation. Since there are 100 observations in the sample, each observation has a weight of 1/100. Thus, the ECDF value at the smallest observation is 1/100.
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Question 3
Write the ordered pair for the post office.
?
Understand the Coordinate Plane-Quiz - Level E
DONE
10-
2987
7-
6543 N
4-
2
1
O
2 3 4 5 6
6 7
Post
Office
Library
8 9 10
A
X
Based on the coordinate plane, the ordered pair for the post office is (9, 9).
What is an ordered pair?In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.
By critically observing the coordinate plane shown in the image attached below, we can logically deduce that the coordinates of the post office would be located in quadrant I at point (9, 9).
Similarly, the the coordinates of the library is also located in quadrant I and it is represented by the ordered pair (6, 2).
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3.1-4.6n-3n+8
who can help me on this
Answer: -7.6n + 11.1
Step-by-step explanation:
We will simplify the given expression by combining like-terms.
Given:
3.1 - 4.6n - 3n + 8
Subtract like-terms:
3.1 - 7.6n + 8
Add like-terms:
11.1 - 7.6n
Answer:
[tex] \sf \: -7.6n + 11.1 [/tex]
Step-by-step explanation:
Now we have to,
→ Simplify the given expression.
The expression is,
→ 3.1 - 4.6n - 3n + 8
Let's simplify the expression,
→ 3.1 - 4.6n - 3n + 8
→ -4.6n - 3n + 3.1 + 8
→ (-4.6n - 3n) + (3.1 + 8)
→ (-7.6n) + (11.1)
→ -7.6n + 11.1
Hence, the answer is -7.6n + 11.1.
sophie bikes s miles per hour. her friend rana is slower. Rana bikes r miles per hour. Yesterday, Rana bikes m miles. How many hours did rana bike yesterday?
Answer:
Step-by-step explanation: m miles
A farmer is painting his silo. A typical can of paint covers 400 squared meters. How many cans of paint will the farmer need to buy in order to paint the entire exterior of the silo?
around 13 jars of paint will the farmer need to buy in order to paint the entire exterior of the silo.
To find the number of jars the rancher required to get, you really want to know the surface area of both the cone and cylinder.
The method for finding the SA of the cone would be area =[tex]3.14 x r^2 + 3.14 x r x sqrt(r^2 + h^2)[/tex].
SA= 1976.0617791 [tex]m^2[/tex].
The method for finding the SA of the cylinder under the cone would be 3.14 x d x (d/2 + h).
A = 3097.6103564 [tex]m^2[/tex]
Then, at that point, you need to add 1976.0617791+3097.6103564 which gives you 5,073.8. (5,073 is the total surface area of the storehouse)
Then, the partition that by 400.
5,073.8/400 = 12.7
He would have to purchase around 13 jars of paint to have the option to paint the whole storehouse.
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Find the volume of these figures. Then describe the patterns you see. Can you determine the volume of the next figure in the pattern? (picture included)
(if you are not sure of the answer please do not answer because someone just got 15 points from me and the answer was not correct.. i think...)
Based οn this pattern, we can determine the vοlume οf the next figure in the pattern by using a height οf 4 units: 1 x 1 x 4 = 4 cubic units.
What is a Cube?A cube is three-dimensiοnal sοlid οbject that has six square faces οf equal size. It is special type οf rectangular prism in which all six faces are squares οf equal size. A cube has twelve edges οf equal length and eight vertices where three edges meet.
Let's analyze each figure separately and find the vοlume using the fοrmula fοr the vοlume οf a rectangular prism: length x width x height.
First figure: The length, width, and height are all 1 unit, sο the vοlume is 1 x 1 x 1 = 1 cubic unit.
Secοnd figure: The length and width are still 1 unit, but the height is nοw 2 units. Sο, the vοlume is 1 x 1 x 2 = 2 cubic units.
Third figure: The length and width are still 1 unit, but the height is nοw 3 units. Sο, the vοlume is 1 x 1 x 3 = 3 cubic units.
Frοm the analysis, we can see that the pattern is that the length and width οf each figure remain cοnstant at 1 unit, while the height increases by 1 unit fοr each successive figure.
Based οn this pattern, we can determine the vοlume οf the next figure in the pattern by using a height οf 4 units: 1 x 1 x 4 = 4 cubic units.
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help please i really appreciate it
The correct statement regarding the translation of the functions f(x) and h(x) is given as follows:
The graph of h is a translation of 4 units left and 7 units down of f(x).
What is a translation?A translation happens when either a figure or a function are moved horizontally or vertically on the coordinate plane.
The four translation rules for functions are defined as follows:
Translation left a units: f(x + a).Translation right a units: f(x - a).Translation up a units: f(x) + a.Translation down a units: f(x) - a.The meaning of each translation from function f(x) to function h(x) is given as follows:
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4x° (2x + 6)° plez m
help me
The simplified expression is 8x² + 24x.
What is the distributive property of multiplication over addition?The distributive property of multiplication over addition is a fundamental property of arithmetic that relates multiplication and addition. It states that when you multiply a number by the sum of two or more numbers, you can first distribute the multiplication over each addend and then perform the addition.
In other words, if a, b, and c are any numbers, then:
a x (b + c) = (a x b) + (a x c)
How to solveTo simplify the expression 4x° (2x + 6)°, we can use the distributive property of multiplication over addition.
4x° (2x + 6)°
= 4x° * 2x° + 4x° * 6° (using distributive property)
= 8x² + 24x (simplifying by multiplying)
Therefore, the simplified expression is 8x² + 24x.
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help me find the missing sides to solve this equation
Step-by-step explanation:
use the function sohcahtoa to solve as all these triangles are right angled
find the area of the triangle 16in,25in
hypotenuse^2 = 16^2 + 25^2
hypotenuse^2 = 256 + 625
hypotenuse^2 = 881
hypotenuse = sqrt(881)
hypotenuse ≈ 29.67 inches
Now that we know the length of the hypotenuse, we can use the 16-inch side and the hypotenuse as the base and height of the triangle, respectively. Plugging these values into the formula, we get:
Area = (16 x 29.67) / 2
Area ≈ 237.36 square inches
Therefore, the area of the triangle is approximately 237.36 square inches.
Help!! I suck at math and I’m failing
Answer:
It's very difficult
maby
60% of all Americans live in cities with population greater than 100,000 people. If 47 Americans are randomly selected, find the probability that a. Exactly 25 of them live in cities with population greater than 100,000 people. b. At most 26 of them live in cities with population greater than 100,000 people. c. At least 30 of them live in cities with population greater than 100,000 people. d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.
0.3406 (approx)
Given that 60% of all Americans live in cities with population greater than 100,000 people. If 47 Americans are randomly selected, we need to find the probability thata. Exactly 25 of them live in cities with population greater than 100,000 people.b. At most 26 of them live in cities with population greater than 100,000 people.c. At least 30 of them live in cities with population greater than 100,000 people.d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.Probability is defined as the ratio of the favorable outcomes to the total outcomes of an event. The formula to calculate probability is given by;Probability = Number of favorable outcomes / Total number of outcomesa. Exactly 25 of them live in cities with a population greater than 100,000 people.Probability of exactly 25 Americans living in cities with population > 100,000 people is given by the probability mass function of the binomial distribution.P( X = 25) = 47 C 25 * (0.6)25 * (0.4)22= 0.1213 (approx)b. At most 26 of them live in cities with population greater than 100,000 people.We need to find the probability of at most 26 Americans living in cities with population greater than 100,000 people. This means the number of Americans living in cities with population greater than 100,000 is 0, 1, 2, ..., 26.P(X ≤ 26) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 26)P(X ≤ 26) = ΣP(X = r)Where r varies from 0 to 26P(X ≤ 26) = Σ (47 C r * (0.6)r * (0.4)47-r )r = 0 to 26= 0.6413 (approx)c. At least 30 of them live in cities with population greater than 100,000 people.We need to find the probability of at least 30 Americans living in cities with population greater than 100,000 people. This means the number of Americans living in cities with population greater than 100,000 is 30, 31, 32, ..., 47.P(X ≥ 30) = P(X = 30) + P(X = 31) + P(X = 32) + ... + P(X = 47)P(X ≥ 30) = ΣP(X = r)Where r varies from 30 to 47P(X ≥ 30) = Σ (47 C r * (0.6)r * (0.4)47-r )r = 30 to 47= 0.0031 (approx)d. Between 23 and 28 (including 23 and 28) of them live in cities with population greater than 100,000 people.We need to find the probability of the number of Americans living in cities with population greater than 100,000 is between 23 and 28 (both inclusive).P(23 ≤ X ≤ 28) = P(X = 23) + P(X = 24) + ... + P(X = 28)P(23 ≤ X ≤ 28) = ΣP(X = r)Where r varies from 23 to 28= Σ (47 C r * (0.6)r * (0.4)47-r )r = 23 to 28= 0.3406 (approx)Hence, the required probabilities are,a. P(X = 25) = 0.1213 (approx)b. P(X ≤ 26) = 0.6413 (approx)c. P(X ≥ 30) = 0.0031 (approx)d. P(23 ≤ X ≤ 28) = 0.3406 (approx)
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