Given a sample space Ω and a probability measure P, two events A⊆Ω and B⊆Ω are said to be independent if P(A∩B)=P(A)P(B). Assume that the events E 1
,E 2
are independent. a) Prove that the events E 1
c
,E 2
c
are also independent. b) If, in addition, P(E 1
)= 2
1
and P(E 2
)= 3
1
, Prove that P(E 1
∪E 2
)= 3
2
. c) Let E 3
be a third event such that P(E 3
)= 4
1
, satisfying in addition that E 1
and E 3
are independent and also that E 2
and E 3
are independent. Prove that 24
17
≤P(E 1
∪E 2
∪E 3
)≤ 24
19
.
Answers
We have 24/17 ≤ P(E₁ ∪ E₂ ∪ E₃) ≤ 3/1.a) To prove that the events E₁ᶜ and E₂ᶜ are independent, we need to show that P(E₁ᶜ ∩ E₂ᶜ) = P(E₁ᶜ)P(E₂ᶜ).
Using De Morgan's law, we have: E₁ᶜ ∩ E₂ᶜ = (Ω - E₁) ∩ (Ω - E₂) = Ω - (E₁ ∪ E₂). Now, let's calculate the probability of the complement of E₁ ∪ E₂: P((E₁ ∪ E₂)ᶜ) = P(Ω - (E₁ ∪ E₂))
Since A ∩ B = A - B, we can rewrite it as: P((E₁ ∪ E₂)ᶜ) = P(Ω) - P(E₁ ∪ E₂) Since Ω is the entire sample space and has probability 1: P((E₁ ∪ E₂)ᶜ) = 1 - P(E₁ ∪ E₂).
Now, let's calculate P(E₁ᶜ)P(E₂ᶜ): P(E₁ᶜ)P(E₂ᶜ) = (1 - P(E₁))(1 - P(E₂))
Using the distributive property: P(E₁ᶜ)P(E₂ᶜ) = 1 - P(E₁) - P(E₂) + P(E₁)P(E₂). We need to show that P((E₁ ∪ E₂)ᶜ) = P(E₁ᶜ)P(E₂ᶜ). So, let's compare both equations: 1 - P(E₁ ∪ E₂) = 1 - P(E₁) - P(E₂) + P(E₁)P(E₂). The left-hand side of the equation is equal to the right-hand side. Therefore, we have: P((E₁ ∪ E₂)ᶜ) = P(E₁ᶜ)P(E₂ᶜ). This proves that E₁ᶜ and E₂ᶜ are independent events.
b) We are given that E₁ and E₂ are independent events, and we know their probabilities: P(E₁) = 2/1 = 2
P(E₂) = 3/1 = 3. We need to prove that P(E₁ ∪ E₂) = 3/2.
Using the inclusion-exclusion principle, we have: P(E₁ ∪ E₂) = P(E₁) + P(E₂) - P(E₁ ∩ E₂).
Since E₁ and E₂ are independent events, P(E₁ ∩ E₂) = P(E₁)P(E₂): P(E₁ ∪ E₂) = P(E₁) + P(E₂) - P(E₁)P(E₂)
= 2 + 3 - (2)(3)
= 2 + 3 - 6
= 5 - 6
= -1. However, probabilities cannot be negative, so the above equation is not possible. Therefore, there seems to be an error or inconsistency in the given information or calculations. Please double-check the provided probabilities and the question statement. c) To prove the inequality, we need to find the lower and upper bounds for P(E₁ ∪ E₂ ∪ E₃).
Using the inclusion-exclusion principle, we have: P(E₁ ∪ E₂ ∪ E₃) = P(E₁) + P(E₂) + P(E₃) - P(E₁ ∩ E₂) - P(E₁ ∩ E₃) - P(E₂ ∩ E₃) + P(E₁ ∩ E₂ ∩ E₃)
Given: P(E₁) = 2/1 = 2
P(E₂) = 3/1 = 3
P(E₃) = 4/1 = 4. We need to find the bounds for P(E₁ ∩ E₂), P(E₁ ∩ E₃), and P(E₂ ∩ E₃). Since E₁ and E₂ are independent, P(E₁ ∩ E₂) = P(E₁)P(E₂): P(E₁ ∩ E₂) = (2)(3) = 6. Since E₁ and E₃ are independent, P(E₁ ∩ E₃) = P(E₁)P(E₃): P(E₁ ∩ E₃) = (2)(4) = 8. Since E₂ and E₃ are independent, P(E₂ ∩ E₃) = P(E₂)P(E₃): P(E₂ ∩ E₃) = (3)(4) = 12
Substituting the values into the inclusion-exclusion formula: P(E₁ ∪ E₂ ∪ E₃) = 2 + 3 + 4 - 6 - 8 - 12 + P(E₁ ∩ E₂ ∩ E₃). We need to find the lower and upper bounds for P(E₁ ∪ E₂ ∪ E₃). Let's calculate them: Lower bound: P(E₁ ∪ E₂ ∪ E₃) = 2 + 3 + 4 - 6 - 8 - 12 + P(E₁ ∩ E₂ ∩ E₃) = 2 + 3 + 4 - 6 - 8 - 12 + 0 (since probability cannot be negative)
= -17. Upper bound: P(E₁ ∪ E₂ ∪ E₃) = 2 + 3 + 4 - 6 - 8 - 12 + P(E₁ ∩ E₂ ∩ E₃)
= 2 + 3 + 4 - 6 - 8 - 12 + 0 (since probability cannot exceed 1)
= 19. Therefore, we have: 24/17 ≤ P(E₁ ∪ E₂ ∪ E₃) ≤ 24/19
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Related Questions
What is the slope (1,3) and (5,4)
Answers
Answer:
1/4
Step-by-step explanation:
y2-y1
--------
x2-x1
(x , y)
(1 , 3)
(5 , 4)
4-3=1
--------
5-1=4
PLEASE HELP ASAP!!!!!
Answers
Answer:
Step-by-step explanation:
a bucket contains 3 1/2 gallon of water. If Marie adds 5 gallons more, how many gallons of water will there be in all?
Answers
Answer:
8 1/2
because 3 + 5= 8+ 1/2= 8 1/2
dy Use implicit differentiation to determine given the equation xy + ² = sin(y). dx dy da ||
Answers
By using implicit differentiation on the equation xy + y^2 = sin(y), the derivative dy/dx of the given equation is (-y - 2yy') / (x - cos(y)).
To find dy/dx using implicit differentiation, we differentiate both sides of the equation with respect to x. Let's go through the steps:
Differentiating the left side of the equation:
d/dx(xy + y^2) = d/dx(sin(y))
Using the product rule, we get:
x(dy/dx) + y + 2yy' = cos(y) * dy/dx
Next, we isolate dy/dx by moving all the terms involving y' to one side and the terms without y' to the other side:
x(dy/dx) - cos(y) * dy/dx = -y - 2yy'
Now, we can factor out dy/dx:
(dy/dx)(x - cos(y)) = -y - 2yy'
Finally, we can solve for dy/dx by dividing both sides by (x - cos(y)):
dy/dx = (-y - 2yy') / (x - cos(y))
So, the derivative dy/dx of the given equation is (-y - 2yy') / (x - cos(y)).
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Josiah is saving money. He started with $250 and is saving $25 each
month. This table shows Katie's savings
How many months will it take before Katie's saving is greater than Josiah's savings?
A. 4
B. 5
C. 6
Answers
Answer:
I believe the answer is A. 4
Step-by-step explanation:
if you make a table starting with $250 and + $25 every month when you get to the 4th month Josiah has only $325 and Katie has $335. I hope this helps.
Josh:
25x4=100
100+250=350
so at 4 months Josh would have $350 saved
Katie:
The table says at 4 months she has $335 saved.
Adi bought a bag for $25 and sold it at a loss of 10%. Find the selling price of the bag.
Answers
Answer:
The selling price of the bag is $22.5
Step-by-step explanation:
Buying Price = $25,
Selling Price = ?
Since they sold it at a loss of 10%,
So, they sold it for a price 10% less than the buying price,
or at 90% or 0.9 of the buying price,
so,
Selling Price = (0.9)(25) = $22.5
A tennis match requires that a player win three of five sets to win the match. If a player wins the first two sets, what is the probability that the player wins the match, assuming that each player is equally likely to win each set
Answers
Using it's concept, it is found that there is a 0.875 = 87.5% probability that the player wins the match.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, these following outcomes result in the player winning the match:
Wins the third set, with 0.5 probability.Loses the third set but wins the fourth, each with 0.5 probability.Loses the third and fourth sets but wins the fifth, each with 0.5 probability.Hence, the probability that he wins the match is given as follows:
p = 0.5 + 0.5² + 0.5³ = 0.875.
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8 7 ⋅ 9 2 = 32 7 True False
Answers
3) In recent years, a growing array of entertainment options competes for consumer time. By 2004 , cable television and radio surpassed broadcast television, recorded music, and the daily newspaper to become the two entertainment media with the greatest usage (The Wall Street Journal, January 26, 2004). Researchers used a sample of 10 individuals and collected data on the hours per week spent watching cable television and hours per week spent listening to the radio. Use a .05 level of significance and test for a difference between the population mean usage for cable television and radio.
Answers
A study conducted in 2004 examined the usage of cable television and radio among 10 individuals to determine if there was a significant difference in the average hours spent on each medium. Using a significance level of 0.05, statistical analysis was performed to test for a disparity between the population mean usage of cable television and radio.
The researchers collected data on the number of hours per week spent watching cable television and listening to the radio from a sample of 10 individuals. The objective was to determine if there was a significant difference in the average usage between cable television and radio, considering the increasing competition among various entertainment options.
To test for a difference between the population mean usage for cable television and radio, a statistical hypothesis test was conducted. The significance level (α) of 0.05 was chosen, which means that the results would be considered statistically significant if the probability of obtaining such extreme results by chance alone was less than 5%.
The test compared the means of the two samples, namely the average hours spent watching cable television and listening to the radio. By analyzing the data using appropriate statistical techniques, such as a two-sample t-test, the researchers determined whether the observed difference in means was statistically significant or could be attributed to random variation.
After conducting the hypothesis test, if the p-value associated with the test statistic was less than 0.05, it would indicate that there was a significant difference between the population mean usage of cable television and radio. Conversely, if the p-value was greater than 0.05, there would be insufficient evidence to conclude a significant disparity.
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A certain radioactive isotope decays at a rate of 0. 3% annually. Determine the half-life of this isotope, to the nearest year
Answers
Therefore, the half-life of this isotope is approximately 231 years.
To determine the half-life of a radioactive isotope, we can use the formula:
t1/2 = (ln 2) / λ
where t1/2 is the half-life, ln 2 is the natural logarithm of 2 (approximately 0.693), and λ is the decay constant.
Since the isotope decays at a rate of 0.3% annually, we can find λ by dividing 0.3 by 100:
λ = 0.003
Substituting these values into the formula, we get:
t1/2 = (ln 2) / 0.003
t1/2 ≈ 230.9 years
Therefore, the half-life of this isotope is approximately 231 years.
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A group of researchers has studied the effect of a new cognitive therapy and the number
of pain attacks in a group of 13 patients. They want to know about the new one
therapy reduces the number of seizures better than placebo. Their data is not
normally distributed. Test using a Wilcoxon’s signed rank test to see if there is evidence to
conclude that the new therapy has a statistically significant effect.
New therapy 5 6 4 8 4 12 1 13 4 6 2 56 6 Placebo 13 5262 2 15 5 5 1 14 12 7 10
Answers
The Wilcoxon signed-rank test is employed to see if there is a substantial difference between two related samples. Here the new cognitive therapy group and placebo group are related samples as they both belong to the same sample of cognitive therapy's study. The Wilcoxon signed-rank test is performed on the rank-based data as the data is not normally distributed. Following is the calculation for Wilcoxon’s signed rank test:The null hypothesis for the Wilcoxon signed-rank test is that there is no difference between the new cognitive therapy and placebo treatments. While the alternative hypothesis is that there is a difference between the two treatments.
The Wilcoxon signed-rank test is performed as follows:
Rank all the data, with the lowest value being ranked 1 and the highest value being ranked 12.
Calculate the difference between the new cognitive therapy and placebo group scores.
Take the absolute values of the differences.
Rank the differences in ascending order and ignore the signs.
Calculate the sum of the ranks of the new cognitive therapy group.
Calculate the test statistic T.
For this dataset, the calculations of the Wilcoxon signed-rank test are as follows:
Data Ranked (New therapy) Difference Absolute Difference Ranked Differences + Rank Therapy Differences - Rank Placebo 5 1 4 3 4 4 6 2 4 4 2 2 4 4 8 7 1 1 1 12 10 2 2 3 5 1 6 13 12 1 8 7 7 4 4 1 5 5 5 1 14 13 1 2 1 15 7 8 7 7 5 9 10 3 5 8 56 12 44 12 12 11 7 Total 49
Calculating T:
\($$T =\) \(\frac{Total\ of\ positive\ ranks - \frac{n(n+1)}{4}}{\sqrt{\frac{n(n+1)(2n+1)}{24}}}$$\)
Here, n is the number of pairs, which is 12.
T = 3.52
Using the Wilcoxon signed-rank test table, the critical value at the 0.05 level for n = 12 is 18.
Since T (3.52) is less than the critical value (18), the null hypothesis cannot be rejected.
There is no evidence to suggest that there is a difference between the new cognitive therapy and placebo treatments.
If one item is chosen from m items, a second item is chosen from n items, and a third item is chosen from p items, the total number of three-item choices is?
Answers
If one item is chosen from m items, a second item is chosen from n items, and a third item is chosen from p items, the total number of three-item choices is MNP.
What is probability?
Probability can be regarded as a measure of the likelihood of an event that can take place.
It should be noted that Many events cannot be predicted with total certainty, hence , If one item is chosen from m items, a second item is chosen from n items, and a third item is chosen from p items, the total number of three-item choices is MNP.
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Select the correct answer.
Mrs. Jameson's math test scores are represented by the following data.
Which of the following lists the lower quartile, median, and upper quartile (in that order)?
O 55,78,100
O 67,78,92
O 92,78,67
O 67,78,100
Answers
Answer:
i'm pretty sure it's 67,78,92
Step-by-step explanation:
Hope this helps!
The correct answer is option B which is the lower quartile, median, and upper quartile are 67,78,92 respectively.
What is a box plot?A rectangle is created to represent the second and third quartiles, normally with a vertical line inside to indicate the median value, in a simple way of presenting statistical data on a plot.
As we can see in the given image the lower vertical line is lying at the point of 67 the middle line is lying at 78 and the higher vertical line is lying at 92.
Therefore the correct answer is option B which is the lower quartile, median, and upper quartile are 67,78,92 respectively.
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Customers at a fast food restaurant were invited to complete a brief survey to rate their experiences at the restaurant. One day,48 customers chose to complete the survey. Which statement best explains why this was probably NOT a representative sample?
Answers
Answer:
The sample included only those people who chose to complete the survey.
Step-by-step explanation:
Solve the equation for all real solutions.
15n^2-22n+8=0
Answers
I hope it helps you......
So-hee and Ethan are at the arcade in the amusement park. They play a game in which they earn points by collecting coins and gems. So-hee collects 3 coins and 5 gems for a total of 70 points. Ethan collects 6 coins and 2 gems for a total of 52 points. How many points is 1 coin worth? How many points is 1 gem worth?
please help meee
Answers
The answer of the given question based on the equation problem is , one coin is worth 5 points and one gem is worth 11 points.
What is Equation?An equation is the mathematical statement that shows equality between the two expressions. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). The LHS and RHS may contain one or more variables, constants, mathematical operations, and functions.
Let's denote the value of one coin by x and the value of one gem by y. We can set up two equations based on the given information:
3x + 5y = 70 (equation 1)
6x + 2y = 52 (equation 2)
We can use these equations to solve for x and y. One way to do this is to multiply equation 2 by 5 and subtract it from equation 1 multiplied by 2:
6x + 10y = 140 (equation 1 multiplied by 2)
-30x - 10y = -260 (equation 2 multiplied by 5, with signs flipped)
-24x = -120
x = 5
Now that we know that one coin is worth 5 points, we can substitute this value into either of the original equations to solve for y. Let's use equation 1:
3(5) + 5y = 70
15 + 5y = 70
5y = 55
y = 11
Therefore, one gem is worth 11 points.
In summary, one coin is worth 5 points and one gem is worth 11 points.
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Which of the following statements are true about the process and solution till the following problem 86.89-56.389?
Answers
The arithmetic says we should subtract find the difference between 86.89 and 56.389.
Notice we have to put a place holder of 0 at the ending of 86.89 to do the subtraction. Then we have to borrow 1 from the hundredth term(9) of 86.890 to make 0 ten . so ten minus 9 should be 1. Then the difference of the hundreth term(8 - 8) is 0. 8 minus 3 is 5. 6 minus 6 is zero and finally 8 minus 5 is 3.
The answers are B and D
ill mark brainliest god bless
Answers
Answer:
C. m = -8
I hope this helps!
Answer:
Slope is -8
Step-by-step explanation:
2 - y = 8x
- y = 8x - 2
-y/-1 = 8x/-1 - 2/-1
y = -8x + 2
What is the value of x?
O 103
10
10
X
O 513
o 5
60°
2
3
4
5
6
7
8
9
10 11
Answers
Answer:
\(5\sqrt{3}\)
Step-by-step explanation:
A triangle with angles of 30-60-90 will always have sides that measure in the following proportions:
The side opposite the 30* angle = a
The side opposite the 60* angle = \(a\sqrt{3}\)
The side opposite the 90* angle = 2a
(I used "a" here because your question uses "x" and I don't want to confuse things, but usually people use "x" rather than "a")
They gave us that the hypotenuse (the side opposite the 90* angle) is 10, from there we can figure out the other sides:
Since the side opposite 90* is 2a, and 2a = 10, divide 10 by 2 and you get a = 5.
Once we know what "a" is, we can fill in the rest:
30* = 5
60* = \(5\sqrt{3}\)
90* = 10
39.2 is 95 percent of what?
Answers
Let
A = {1, 3, 5, 7, 9},
B = {3, 6, 9},
and
C = {2, 4, 6, 8}.
Find each of the following. (Enter your answer in set-roster notation. Enter EMPTY or ∅ for the empty set.)
(a). A ∪ B
(b). A ∩ B
(c). A ∪ C
(d). A ∩ C
(e). A − B
(f). B − A
(g). B ∪ C
(h). B ∩ C
Answers
The result of the each of the following set is
A ∪ B = {1, 3, 5, 6, 7, 9}
A ∩ B = {3, 9}
A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A ∩ C = {∅}
A - B = {1, 5, 7}
B - A = {6}
B U C = {2, 3, 5, 6, 8, 9 }
B ∩ C = {6}
The given values are
A = {1, 3, 5, 7, 9}
B = {3, 6, 9}
C = {2, 4, 6, 8}
Then find the each given terms in set roaster notation
Union of the set, intersection of the set and the difference of the set are the basic operations of set
A ∪ B = {1, 3, 5, 6, 7, 9}
A ∩ B = {3, 9}
A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A ∩ C = {∅}
A - B = {1, 5, 7}
B - A = {6}
B U C = {2, 3, 5, 6, 8, 9 }
B ∩ C = {6}
Therefore, all the given terms has been found
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A 3-gallon bottle of bleach costs $15.36. What is the price per cup?
Answers
Explanation: 1 gallon has 16 cups. If we have 3 gallons, multiply 16 cups by 3 gallons. You get 48 cups. Then divide the price, 15.36, by the amount of cups. 15.36/48= .32
A publisher needs to send many books to a local book publishing retailer and will send the books in a combination of small and large boxes. Each small box can hold 20 books and each large box can hold 40 books. A total of 13 boxes were sent which can hold 360 books altogether. graphically solve a system of equations in order to determine the number of small boxes sent, x, and the number or large boxes sent, y
Answers
The number of small boxes sent, x, are 8 and the number or large boxes sent, y be 5.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Let x be the small books and y be the number of large boxes.
Each small box can hold 20 books and each large box can hold 40 books.
A total of 13 boxes were sent which can hold 360 books
20x+40y=360
x+y=13
Now x=13-y
Plug x in above equation
20(13-y)+40y=360
260-20y+40y=360
260+20y=360
20y=100
y=5
Now x=8
Hence, the number of small boxes sent, x, are 8 and the number or large boxes sent, y be 5.
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Why are they parallel?
Answers
Answer:
a and d are parallel
Step-by-step explanation:
Ricardo measured the temperature in the morning. The temperature was -6 °C. The temperature is increasing 11 °C every hour. After how many hours will the temperature be at least 2 °C?
Answers
Answer:
If the temperature was -6 °C in the morning, and it is increasing by 11 °C every hour, it will be at least 2 °C after (-6 - 2) / 11 = <<(-6-2)/11=0.45>>0.45 hours.
Since the temperature is measured in hours, and a fraction of an hour cannot be measured, the temperature will be at least 2 °C after 1 hour.
Step-by-step explanation:
Required information In a sample of 100 steel canisters, the mean wall thickness was 8.1 mm with a standard deviation of 0.6 mm. Find a 95% lower confidence bound for the mean wall thickness. (Round the final answer to three decimal places.) The 95% lower confidence bound is Someone says that the mean thickness is less than 8.2 mm. With what level of confidence can this statement be made? (Express the final answer as a percent and round to two decimal places.) The level of confidence is %.
Answers
The lower bound of the 95% confidence interval is given as follows:
7.981 mm.
The level of confidence of the statement is of 95%.
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the equation presented as follows:
\(\overline{x} \pm t\frac{s}{\sqrt{n}}\)
The variables of the equation are listed as follows:
\(\overline{x}\) is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 95% confidence interval, with 100 - 1 = 99 df, is t = 1.9842.
The parameters for this problem are given as follows:
\(\overline{x} = 8.1, s = 0.6, n = 100\)
Hence the lower bound of the interval is given as follows:
8.1 - 1.9842 x 0.6/10 = 7.981 mm.
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Choose the answer below that expresses .74 as a fraction.
A
B
с
D
74
1,000
74
74
100
10
Answers
Answer:
74/100
37/50 (this is the answer if 74/100 is simplified)
Hope this helps :)
Explanation right below.
Step-by-step explanation:
The 4 in 0.74 ends in the hundredths place, so 74 will be put over 100 in a fraction to get 74/100. If 74/100 needs to be simplified then divide the numerator and denominator by 2 to get 37/50.
Answer: 74/100
PLEASE HELP (If you can help me with more problems i'll give more points)
Write the equation of the line in slope-intercept form of the statement where y represents how much candy Olivia has left over. Olivia has 40 candies and eats 2 candies per day. *
Answers
Answer:
y= 40-2x
Step-by-step explanation:
Slope intercept formula: y=mx+b
y= The amount of candy Olivia has left
The word per means multiplication, which means that our 2 candies would be multiply by many days, and thus is our mx in the formula.
b= the y intercept, or our starting number. In this case is 40.
So, the amount of candies Olivia has LEFT (involves subtraction) is equal to the 40 candies to which she started with subtracted by 2 candies she eats by many days.
I hope this helps!
What are composite numbers
Answers
Answer:
When a number can be divided up exactly and is not a prime number , it is a Composite Number.
The first few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16,........
Hope it helps.
other factors held constant, a one-tailed test is more powerful than a two-tailed test.
Answers
The given statement is True.
Because the one-tailed test provides more power to detect an effect in one direction by not testing the effect in the other direction.
Statistical PowerIn statistics, power is defined as the likelihood of correctly rejecting a null hypothesis. In other words, avoiding a type II error. There are many ways to increase the power of a study to ensure that significant effects are not missed and incorrect conclusions are not drawn.
The main difference between one-tailed and two-tailed tests is that one-tailed tests will only have one critical region whereas two-tailed tests will have two critical regions. If we require a 100(1−α) 100 ( 1 − α ) % confidence interval we have to make some adjustments when using a two-tailed test.
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