What is .068 as a percentage

The probability of getting at least one balloon of each of the 30 types in a set of 50 balloons is approximately 95.55%.

The probability of getting at least one balloon of each kind out of the 30 available types can be calculated using the Principle of Inclusion-Exclusion.

First, the probability of getting one specific type of balloon out of 30 is 1/30.

The probability of not getting that specific type of balloon is 29/30.

Thus, the probability of getting at least one of that specific type of balloon in a set of 50 balloons is:

P(getting at least one of that specific type of balloon) = 1 - P(not getting that specific type of balloon)
P(getting at least one of that specific type of balloon) = 1 - (29/30)^50

Now, we need to consider all 30 types of balloons. Using the Principle of Inclusion-Exclusion, the probability of getting at least one balloon of each type is:

P(getting at least one of each type) = P(getting at least one of the first type) ∩ P(getting at least one of the second type) ∩ ... ∩ P(getting at least one of the thirtieth type)

P(getting at least one of each type) = 1 - P(not getting at least one of any type)
P(getting at least one of each type) = 1 - [(29/30)^50]^30

P(getting at least one of each type) = 1 - 0.0445

P(getting at least one of each type) = 0.9555 or 95.55%

Therefore, the probability of getting at least one balloon of each of the 30 types in a set of 50 balloons is approximately 95.55%.

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The experimental probability of the coin landing heads up is calculated by dividing the number of times the coin landed heads up (16) by the total number of flips (40). So the experimental probability of the coin landing heads up is:

P(heads up) = 16/40

Simplifying the fraction by dividing both the numerator and denominator by 8, we get:

P(heads up) = 2/5 or 0.4

Therefore, based on the results, the experimental probability of the coin landing heads up is 0.4 or 2/5.

To find the experimental probability of the coin landing heads up, you'll need to use the following formula:

Experimental probability = (Number of successful outcomes) / (Total number of trials)

In this case, the successful outcome is the coin landing heads up, which occurred 16 times. The total number of trials is 40 flips. So, the experimental probability would be:

Experimental probability (heads up) = (16 successful outcomes) / (40 total flips)

Now, divide 16 by 40 to get the probability:

Experimental probability (heads up) = 16/40 = 0.4 or 40%

So, based on the results, the experimental probability of the coin landing heads up is 40%.

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In any case, it's important to formulate a hypothesis that is testable and based on prior knowledge or observations. This ensures that the experiment is designed to answer a specific question or test a specific idea, and that the results are meaningful and informative.

In the pillbug experiment, the hypothesis formulated was related to the food preferences of the pillbugs.

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The absolute increase is 900 people, and the relative (percent) increase is 27.27%.

We will first find the absolute increase and then the relative (percent) increase.
Absolute increase:
Subtract the initial population from the final population: 4200 (2009 population) - 3300 (2006 population)
Calculate the absolute increase: 4200 - 3300 = 900
Absolute increase:

900 people
Relative (percent) increase:
Calculate the absolute increase (which we found earlier): 900 people.

Divide the absolute increase by the initial population: 900 / 3300
Multiply the result by 100 to find the percentage: (900 / 3300) * 100
Calculate the relative (percent) increase: (900 / 3300) * 100 = 27.27%
Relative (percent) increase: 27.27%.

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Tristan flies the plane at a speed of 493 mph when there is no wind.

To solve this problem, we can use the formula:

distance = rate x time

Let's start by finding Tristan's speed when flying against the headwind. We know that the distance he covers is 3795 miles and the wind speed is 7 mph. Let's assume his speed without the wind is x mph. Then his speed against the wind is (x - 7) mph. Using the formula, we can write:

3795 = (x - 7) * t1

where t1 is the time it takes to fly 3795 miles against the headwind.

Next, let's find his speed when flying with the wind. We know that the distance he covers is the same (3795 miles), but this time it takes him 14 hours less. So the time it takes to fly with the wind is t1 - 14. His speed with the wind is (x + 7) mph. Using the formula again, we can write:

3795 = (x + 7) * (t1 - 14)

Now we have two equations with two unknowns (x and t1). We can solve for x by eliminating t1. Let's start by expanding the second equation:

3795 = x*t1 + 7*t1 - 14x - 98

Add 14x to both sides:

14x + 3795 = x*t1 + 7*t1 - 98

Add 98 to both sides:

14x + 3893 = x*t1 + 7*t1

Substitute t1 from the first equation:

14x + 3893 = (x - 7)*t1 + 7*t1

14x + 3893 = x*t1 - 7*t1 + 7*t1

14x + 3893 = x*t1

Now we have a single equation with x and t1. We can substitute t1 from the first equation again:

14x + 3893 = x * (3795/(x-7))

Simplify by multiplying both sides by (x-7):

14x(x-7) + 3893(x-7) = 3795x

Expand and simplify:

14x^2 - 98x + 3893x - 27251 = 3795x

Subtract 3795x from both sides:

14x^2 - 6722x - 27251 = 0

Now we can solve for x using the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac))/(2a)

where a = 14, b = -6722, and c = -27251. Plugging in the values, we get:

x = (6722 +/- sqrt(6722^2 + 4*14*27251))/(2*14)

x = (6722 +/- sqrt(45619268))/28

x = (6722 +/- 6742)/28

So x can be either 481 or 493. We need to choose the positive value because Tristan's speed can't be negative. Therefore, the answer is: 493.

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The arrival time of an elevator in a 12-story dormitory is equally likely at any time range during the next 4.6 minutes. The probability that the wait for an elevator is more than 3.5 minutes is 0.239.

a. Expected arrival time:
Since the elevator is equally likely to arrive at any time during the next 4.6 minutes, the expected arrival time will be the midpoint of this time range.
Expected arrival time = (0 + 4.6) / 2 = 2.30 minutes
b. Probability of arrival in less than 3.5 minutes:
To calculate this probability, we need to find the proportion of the time range (4.6 minutes) that is less than 3.5 minutes.
Probability = (3.5 minutes) / (4.6 minutes) = 0.7609 (rounded to 4 decimal places)
Rounded to 3 decimal places, the probability is 0.761.
c. Probability of waiting more than 3.5 minutes:
This is the complement of the probability calculated in part b. We can find it by subtracting the probability of arrival in less than 3.5 minutes from 1.
Probability = 1 - 0.7609 = 0.2391 (rounded to 4 decimal places)
Rounded to 3 decimal places, the probability is 0.239.

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The number of permutations in factorial form represents the number of ways to arrange a set of objects without repetition. The formula for permutations is n!, where n is the number of objects.

In this case, you and six of your friends need to sit in the back seat of a limousine. Since the order of seating matters (e.g., the seating arrangement "ABCDEF" is different from "FEDCBA"), we can use the permutation formula to calculate the number of different ways:

Number of permutations = 7!

Let's simplify this expression:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

  = 5040

Therefore, there are 5,040 different ways for you and your six friends to sit in the back seat of the limousine.

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The probability that Bill will choose apple pie and Laura will choose strawberry cheesecake for dessert is [tex]\frac{1}{16}[/tex].

You want to know the probability that Bill will choose apple pie and Laura will choose strawberry cheesecake for dessert.

Step 1: Determine the probability of each individual event.
Since there are 4 options on the dessert menu, the probability of Bill choosing apple pie is [tex]\frac{1}{4}[/tex], and the probability of Laura choosing strawberry cheesecake is also [tex]\frac{1}{4}[/tex].

Step 2: Calculate the joint probability of both events happening.
To find the probability of both events happening, multiply the individual probabilities: [tex](\frac{1}{4}) (\frac{1}{4}) = \frac{1}{16}[/tex]

So, the probability that Bill will choose apple pie and Laura will choose strawberry cheesecake for dessert is [tex]\frac{1}{6}[/tex].

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The other names for the Fundamental Theorems of Calculus are the Integral Evaluation Theorem and the Fundamental Theorem of Calculus, Part One and Part Two.

The Fundamental Theorem of Calculus is a significant concept in calculus that connects integration and differentiation. It essentially states that integration and differentiation are inverse operations of each other. The theorem has two parts: Part One and Part Two.

Part One of the Fundamental Theorem of Calculus states that if a function f(x) is continuous on the interval [a,b], then the definite integral of f(x) from a to b can be evaluated using an antiderivative of f(x) at the endpoints a and b.

Part Two of the Fundamental Theorem of Calculus, also known as the Integral Evaluation Theorem, extends the concept of Part One by stating that if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b can be evaluated as the difference between the antiderivative evaluated at the endpoints a and b. This theorem is often used to evaluate definite integrals.

Therefore, the other names for the Fundamental Theorems of Calculus are the Integral Evaluation Theorem and the Fundamental Theorem of Calculus, Part One and Part Two.

These theorems are essential tools in calculus and are used to solve a wide range of problems in many areas of mathematics and science. Understanding and applying these theorems can help to simplify complex problems and enable accurate calculations of integrals.

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The commonly accepted rule of thumb is that a sample size of at least 30 is needed to fulfill the assumption of parametric statistics that the sample is large enough to represent the population. Therefore, the answer is d. 30.

The sample size required to fulfill the assumption of parametric statistics that the sample is large enough to represent the population depends on several factors, including the variability of the population, the level of confidence desired, and the precision required.

However, a commonly cited rule of thumb in statistics is that a sample size of at least 30 is necessary to use parametric statistical methods, such as t-tests and ANOVA. This guideline is based on the central limit theorem, which states that as the sample size increases, the distribution of sample means becomes approximately normal, regardless of the underlying distribution of the population. This means that with a large enough sample size, the distribution of the sample mean is more likely to be a good representation of the population mean.

It is important to note that this guideline may not be appropriate for all situations. For example, if the population is highly variable, a larger sample size may be necessary to accurately represent it. Additionally, if the data is not normally distributed, non-parametric statistical methods may be more appropriate, and the sample size requirement may be different.

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It takes Steve 12 hours to process the research paper alone.

Let's use the terms "research," "hours," and "paper" in our answer.

Step 1: Represent the rate of work for Alan and Steve using variables.
Let A = Alan's rate of work (paper per hour) and S = Steve's rate of work (paper per hour).

Step 2: Set up equations based on the given information.
Alan can complete the research paper in 6 hours, so his rate is 1/6 paper per hour: A = 1/6.
Together, Alan and Steve can complete the paper in 4 hours, so their combined rate is 1/4 paper per hour: A + S = 1/4.

Step 3: Substitute the known value of A (Alan's rate) into the equation and solve for S (Steve's rate).
(1/6) + S = 1/4

Step 4: Solve for S.
To do this, first find a common denominator for the fractions, which is 12. Then, rewrite the equation with equivalent fractions:
(2/12) + S = (3/12)

Now, subtract 2/12 from both sides of the equation:
S = (3/12) - (2/12)

This simplifies to:
S = 1/12

Step 5: Determine how long it takes Steve to complete the research paper alone.
Since Steve's rate is 1/12 paper per hour, it takes him 12 hours to complete the research paper alone.

Answer: It takes Steve 12 hours to process the research paper alone.

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The population parameters that describe the y-intercept and slope of the line relating y and x, respectively, are the true values that would be obtained if the entire population were measured.

The population parameters that describe the y-intercept and slope of the line relating y and x, respectively, are the true values that would be obtained if the entire population were measured. The y-intercept is the value of y when x equals 0, and it represents the starting point of the line. The slope represents the change in y for every one unit change in x, and it determines the steepness of the line. To estimate these population parameters, we use sample statistics such as the sample mean and sample standard deviation. The sample y-intercept and slope are calculated using regression analysis, which involves fitting a line to the data points in order to determine the relationship between x and y. It is important to note that the sample statistics may not be equal to the population parameters, as there is always some degree of error and variability in data. However, by using statistical inference techniques such as confidence intervals and hypothesis testing, we can make inferences about the population parameters based on the sample data. In summary, the population parameters that describe the y-intercept and slope of the line relating y and x, respectively, are the true values that would be obtained if the entire population were measured. These parameters can be estimated using sample statistics and statistical inference techniques.

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Answer:

If Megan can type 84 words in 2 minutes, he can type 42 words in 1 minute. Therefore, it would take Megan 10 minutes to type a 420 word essay.

It takes Megan 10 minutes to type the 420 word essay.

Given that Megan takes 2 minutes to type 84 words.

To find out how many words Megan types in 1 minute, we can divide the 84 words by 2 minutes = [tex]\frac{84}{2}[/tex]  = 42

From the above line, we know that Megan types 42 words in 1 minute. Now, to find out the time taken for Megan to type a 420 word essay, we can divide the 420 by 42 to obtain the time in minutes.

So, time taken = [tex]\frac{420}{42}[/tex] = 10 minutes.

From the above explanation, we can conclude that Megan can type 420 word essay in 10 minutes.

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The  exact value of sin(a/2) is sqrt((1 - sqrt(1 - (21/29)^2)) / 2).

Since sin(a) is positive and a is in quadrant I, we know that cosine of a is also positive. We can use the Pythagorean identity to find cos(a):

cos^2(a) + sin^2(a) = 1

cos^2(a) + (21/29)^2 = 1

cos^2(a) = 1 - (21/29)^2

cos(a) = +/- sqrt(1 - (21/29)^2)

Since a is in quadrant I, we take the positive square root:

cos(a) = sqrt(1 - (21/29)^2)

Next, we can use the tangent identity to find tan(a):

tan(a) = sin(a) / cos(a)

tan(a) = (21/29) / sqrt(1 - (21/29)^2)

Finally, we can use the tangent half-angle formula to find sin(a/2):

sin(a/2) = sqrt((1 - cos(a)) / 2)

sin(a/2) = sqrt((1 - sqrt(1 - (21/29)^2)) / 2)

Therefore, the exact value of sin(a/2) is sqrt((1 - sqrt(1 - (21/29)^2)) / 2).

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There are 29 ways to make a line of 6 marbles using white and black marbles if 2 white marbles cannot be touching.

To solve this problem, we can use the concept of combinations.

First, let's consider the total number of ways to make a line of 6 marbles using white and black marbles without any restrictions. For each of the 6 marbles, we have 2 choices (white or black), so the total number of possible combinations is 2^6 = 64.

Now, let's consider the restriction that 2 white marbles cannot be touching. We can approach this by breaking it down into cases:

Case 1: There are no white marbles in the line.
In this case, we can only use black marbles, so there is only 1 possible combination.

Case 2: There is exactly 1 white marble in the line.
In this case, we can choose any of the 6 positions for the white marble, and then fill the remaining 5 positions with black marbles. So there are 6 possible combinations.

Case 3: There are exactly 2 white marbles in the line, with at least 1 black marble between them.
In this case, we can choose any 2 of the 5 positions between the end white marbles to place the second white marble, and then fill the remaining positions with black marbles. There are 4 possible positions for the second white marble (e.g. WWBWBW, WBWBBW, WBBWBW, WBWBWW), so there are 4*5 = 20 possible combinations.

Case 4: There are exactly 2 white marbles in the line, with no black marbles between them.
In this case, the 2 white marbles must be at the ends of the line (e.g. WWBBBB, BBBBWW). So there are only 2 possible combinations.

Putting it all together, the total number of possible combinations that meet the restriction is 1 + 6 + 20 + 2 = 29. Therefore, there are 29 ways to make a line of 6 marbles using white and black marbles if 2 white marbles cannot be touching.

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Maria's Trattoria can make 128 different pizza combinations using the available toppings, in addition to the plain cheese pizza.

To calculate the number of different pizzas that can be made at Maria's Trattoria, we need to use the concept of permutations and combinations. Permutation is a way of arranging objects in a specific order, whereas combination is a way of selecting objects without considering their order.

In this case, we need to use combinations as the order of toppings doesn't matter. We can select any number of toppings from the given list, including none or all, to create a pizza. So, we need to find the sum of all possible combinations of toppings.

The formula to calculate the number of combinations is nCr = n!/r!(n-r)!, where n is the total number of items, and r is the number of items to be selected.

In this case, there are seven toppings available, and we need to find the number of combinations possible with those seven toppings. Therefore, the number of combinations is:

7C0 + 7C1 + 7C2 + 7C3 + 7C4 + 7C5 + 7C6 + 7C7
= 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1
= 128

So, there are 128 different pizzas that can be made at Maria's Trattoria with the given toppings.

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The net change of the function f(t) = t^2 - 3t between t = 4 and t = 4 + h is h^2 + 5h, and the average rate of change over this same interval is h + 5.

The net change of a function is the overall change in its output value over a given interval. In this case, we are given the function f(t) = t^2 - 3t and asked to determine the net change and average rate of change between t = 4 and t = 4 + h.
To find the net change, we need to evaluate the function at the two endpoints and subtract the smaller value from the larger value. Thus, we have:
f(4 + h) - f(4) = [(4 + h)^2 - 3(4 + h)] - [4^2 - 3(4)]
= [16 + 8h + h^2 - 12 - 3h] - [16 - 12]
= h^2 + 5h
Therefore, the net change of the function between t = 4 and t = 4 + h is given by h^2 + 5h.
Next, we need to find the average rate of change of the function over this same interval. The average rate of change is the slope of the line connecting the two endpoints of the interval. We can find this slope by using the formula:
average rate of change = (f(4 + h) - f(4)) / h
Plugging in the expression for f(t), we get:
average rate of change = [(4 + h)^2 - 3(4 + h) - (4^2 - 3(4))] / h
= (h^2 + 5h) / h
= h + 5
Therefore, the average rate of change of the function between t = 4 and t = 4 + h is given by h + 5.
In summary, the net change of the function f(t) = t^2 - 3t between t = 4 and t = 4 + h is h^2 + 5h, and the average rate of change over this same interval is h + 5.

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Alan can put the 11 books on the shelf, with The Iliad in the first spot and The Odyssey in the last spot, in 9! (362,880) ways

To determine the number of ways Alan can put the 11 books on his bookshelf with The Iliad in the first spot and The Odyssey in the last spot, we can follow these steps:

1. There are 11 spots on the bookshelf, but since The Iliad is in the first spot and The Odyssey is in the last spot, we are left with 9 spots for the remaining 9 books.
2. To arrange the 9 books, we can use the concept of permutations, which refers to the number of ways the books can be ordered.
3. The number of permutations for 9 books is calculated as 9! (9 factorial), which means 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

Therefore, Alan can put the 11 books on the shelf, with The Iliad in the first spot and The Odyssey in the last spot, in 9! (362,880) ways.

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Answer:

Outcome A: 12/25 or 0.48

Outcome B: 9/25 or 0.36

Outcome C: 4/25 or 0.16

Experimental probability distribution:

A: 0.48

B: 0.36

C: 0.16

Experimental design involves manipulating variables to observe their effect on an outcome, while correlational design observes the relationship between variables without manipulation.

In an experimental design, the researcher controls the independent variable(s) and randomly assigns participants to different levels of the independent variable(s) to observe the effect on the dependent variable. This allows for conclusions about cause-and-effect relationships between variables.

In contrast, in a correlational design, the researcher measures two or more variables to observe their relationship without manipulating them. Correlational studies can be used to describe the strength and direction of a relationship between variables, but they cannot establish causality.

While correlational studies are useful for identifying associations between variables, experimental designs are considered the gold standard for establishing causal relationships.

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The required values of the given scale images are as follows:

a. x = 17.5, b. x = 16.67, and x = 5.

a. As we know that scale image is defined as a ratio that represents the relationship between the shape and size of a figure and the corresponding dimensions of the actual figure or object.

As per the given figure a, we can be written as:

35/x = 18/9

35 × 9 = 18x

x = (35 × 9)/18

x = 17.5

b. As per the given figure b, we can be written as:

x/10 = 15/9

x = (15 × 10)/9

x = 150/9

x = 16.67

c. As per the given figure c, we can be written as:

x/2 = 15/6

x = (15 × 2)/6

x = 5

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The absolute maximum is -2 at x = 4, and the absolute minimum is -9 at x = -3.

To find the absolute extrema of f(x) on the interval [-3, 4), we need to first find the critical points and endpoints of the function. The critical points are the points where the derivative of the function is equal to 0 or undefined.

1. Find the derivative of f(x): f'(x) = 1

Since the derivative is a constant, there are no critical points.

2. Evaluate the function at the endpoints of the interval:

f(-3) = -3 - 6 = -9
f(4) = 4 - 6 = -2

3. Compare the values to determine the maximum and minimum:

The maximum value of f(x) on the interval is -2 at x = 4: f(4) = -2.
The minimum value of f(x) on the interval is -9 at x = -3: f(-3) = -9.

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The inequality representing the constraints defined by the Company who manufactures two products is given by x ≤ 2y + 500 , 35x + 50y > $22,500.

'x' is the unit which represents the product A sold.

'y'  is the unit which represents the product B sold.

x is at most 500 units more than twice the number of units of product y.

This situation is represented by inequality,

x ≤ 2y + 500

Second situation is represented as,

Company's profit = $22,500

Square of the company's profit is equal to the sum of 35 times the product A unit sold  50 times product B unit sold.

This implies,

35 × x + 50 × y  > 22,500

⇒ 35x + 50y > $22,500

Therefore, the inequality representing the situation is equal to x ≤ 2y + 500 , 35x + 50y > $22,500.

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Yes, the conditions for a binomial setting have been met in this scenario. A binomial setting requires the following conditions to be met:

1. The experiment consists of a fixed number of identical trials.
2. Each trial results in one of two outcomes: success or failure.
3. The probability of success is constant for each trial.
4. The trials are independent.

In this scenario, we have a fixed number of trials, which is 10. Each trial can result in either a 2 or a non-2, which meets the requirement of two possible outcomes. The probability of rolling a 2 is constant for each trial since the number cube is fair. Finally, each roll of the number cube is independent of the others, so the fourth requirement is met as well. Therefore, we can conclude that the conditions for a binomial setting have been met, and we can use the binomial distribution to calculate the probability of Sarah rolling a certain number of 2's in her 10 rolls of the number cube.

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The standard error of the mean (SEM) is used as the denominator in the

equation for the z-value in a one-sample Z-test.

The formula for the one-sample Z-test is:

z = (sample mean - population mean) / (SEM)

The standard error of the mean (SEM) is used as the denominator in the

equation for the z-value in a one-sample Z-test.

The SEM represents the standard deviation of the sampling distribution of

the mean, which is the distribution of sample means if repeated samples

were taken from the same population. The SEM quantifies the amount of

error that can be expected in the sample mean due to random sampling

variability, and is calculated by dividing the population standard deviation

by the square root of the sample size.

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The least number of people required for a 5% chance of having at least one shared birthday is 15.

To find the least number of people required to have a 5% chance that two of them share the same birthday, we'll use the Birthday Paradox formula:
P(at least 1 shared birthday) = 1 - P(no shared birthdays)
First, let's find the probability of no shared birthdays:
P(no shared birthdays) = (365/365) × (364/365) × (363/365) ×... × (365-n+1)/365
Here, n represents the number of people. Now, we want to find the least n such that:
P(at least 1 shared birthday) ≥ 0.05
Which means:
1 - P(no shared birthdays) ≥ 0.05
We can calculate the probability of no shared birthdays iteratively, starting with n = 2:
1. P(no shared birthdays) = (365/365) × (364/365) = 0.9973
2. P(at least 1 shared birthday) = 1 - 0.9973 = 0.0027
The probability is still less than 0.05, so we increase n to 3:
1. P(no shared birthdays) = (365/365) × (364/365) × (363/365) = 0.9918
2. P(at least 1 shared birthday) = 1 - 0.9918 = 0.0082
Continue this process, increasing n until the probability is greater than or equal to 0.05. After calculating, you'll find that the least number of people required is 14:
1. P(no shared birthdays) = (365/365) × (364/365) × ... × (352/365) ≈ 0.9511
2. P(at least 1 shared birthday) = 1 - 0.9511 ≈ 0.0489
When n = 15:
1. P(no shared birthdays) = (365/365) × (364/365) × ... × (351/365) ≈ 0.9431
2. P(at least 1 shared birthday) = 1 - 0.9431 ≈ 0.0569

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The probability that there are 5 remaining balls is 1/10.

Let's assume that there are initially w white balls and b black balls in the urn. Without loss of generality, let's assume that the first ball drawn is white.

Case 1: All remaining balls are white.

If there are w white balls initially, then the probability of drawing a white ball on the first draw is w / (w + b).

The probability of drawing another white ball on the second draw is (w - 1) / (w + b - 1), and so on.

We need to continue drawing balls until all remaining balls are white. The probability of this happening is:

P1 = w / (w + b) (w - 1) / (w + b - 1) ...  1 / (w + b - w + 1)

Simplifying this expression, we get:

P1 = w! x b! / (w + b)!

Case 2: All remaining balls are black.

If there are b black balls initially, then the probability of drawing a white ball on the first draw is b / (w + b).

The probability of drawing another black ball on the second draw is (b - 1) / (w + b - 1), and so on.

We need to continue drawing balls until all remaining balls are black. The probability of this happening is:

P2 = b / (w + b) (b - 1) / (w + b - 1) ...  1 / (w + b - b + 1)

Simplifying this expression, we get:

P2 = w! x b! / (w + b)!

The probability that there are 5 remaining balls is the sum of P1 and P2, when w + b = 6:

P = P1 + P2 = 3! 3! / 6! + 3! 3! / 6!

  = 2 3! 3! / 6!

  = 1/10

Therefore, The probability that there are 5 remaining balls is 1/10.

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