Let K be the splitting field of x^3 - 5 over Q. • (a) Show that K = Q(∛5, i√3) • (b) Explicitly describe the elements of Aut(K) • (c) Determine what group we have seen before that Aut(K) is isomorphic to • (d) Draw a subgroup diagram for Aut(K) and subfield diagram of K and indicate what subfields correspond to what subgroups under Galois correspondence.
Answers
The splitting field of
x^3 - 5 over Q is K = Q(∛5, i√3)
, which has degree 6 over Q, and so Aut(K) is isomorphic to a subgroup of S6 by the Galois correspondence. The automorphisms that fix
Q(∛5)
form a group of order 2, which is isomorphic to Z2. Since Aut(K) has order 6, it is isomorphic to S3, the group of all permutations of 3 objects.(d) Draw a subgroup diagram for Aut(K) and subfield diagram of K and indicate what subfields correspond to what subgroups under Galois correspondence:
Let K be the splitting field of
x^3 - 5 over Q. (a) Show that
K
= Q(∛5, i√3):
Subfield diagram of K:The subfields that correspond to the subgroups are:
Q corresponds to {id}
Q(i√3) corresponds to {id, τ}
Q(∛5) corresponds to {id, σ}
Q(∜5) corresponds to {id, σ^2}
Q(i√3∛5) corresponds to {id, τσ^2}
Q(i√3∜5) corresponds to {id, τσ}
Q(∛5, i√3) corresponds to Aut
(K)
= {id, σ, σ^2, τσ, τσ^2, τσ^2σ},
which is isomorphic to S3.First, we find that
Q(∛5, i√3)
is a splitting field of x^3 - 5 over Q. Using the tower rule we have
Q ⊂ Q(i√3) ⊂ Q(∛5, i√3)
.Thus, it is sufficient to prove that
[Q(∛5, i√3):Q]
= 6,
and then
Q(∛5, i√3)
must be the splitting field. Since
[Q(∛5):Q]
= 4,
we have
[Q(∛5, i√3):Q(∛5)] ≤ 2,
so it suffices to prove that
[Q(∛5, i√3):Q(i√3)]
= 3.
Consider the irreducible polynomial
x^2 + 3 over Q.
Since it has no roots in Q, it remains irreducible over Q(i√3). Also,
x^3 - 5
= (x - ∛5)(x^2 + ∛5x + ∛5^2)
has no roots in Q(∛5) since ∛5 is not real. Therefore, the splitting field of
x^3 - 5 over Q(i√3)
must be
Q(∛5, i√3).
This splitting field has degree
3 over Q(i√3) since ∛5 is a root of x^3 - 5
and does not lie in Q(i√3).Therefore,
[Q(∛5, i√3):Q]
= [Q(∛5, i√3):Q(i√3)][Q(i√3):Q]
= 3 x 2
= 6, so Q(∛5, i√3)
is the splitting field. Thus,
K
= Q(∛5, i√3).(b)
Explicitly describe the elements of Aut(K):The only non-trivial automorphism of
Q(∛5) over Q sends
∛5 to i√3∛5
since the other choice,
-i√3∛5,
is not real. The other two automorphisms of
Q(∛5, i√3) over Q(∛5)
just fix i√3. Therefore, Aut(K) consists of three elements: the identity, which fixes all of K; the automorphism that sends ∛5 to i√3∛5
and fixes i√3; and the automorphism that sends ∛5 to its conjugate under the complex conjugation and sends i√3 to its conjugate under complex conjugation.(c) Determine what group we have seen before that Aut(K) is isomorphic to:
To know more about Galois correspondence visit:
https://brainly.com/question/31857336
#SPJ11
Related Questions
Imagine you are testing for the effects of two experimental drugs (data set B and C), relative to a control group (Data set A) on a physiological variable. Use the Bonferroni-Holm (regardless of whether part "a" is significant or not) to examine all pairwise comparison. Show all calculations and state your conclusions.
Note: I’ve already added 0.1 and 0.2 to necessary data sets. I’ve completed part a, I need help with part B.
Please show all the steps to solving this, thank you.
Answers
To use the Bonferroni-Holm correction for pairwise comparisons between three groups (A, B, and C), we must adjust the p-value threshold to account for multiple comparisons. First, we calculate the p-value for each pairwise comparison. Then, we rank the p-values from smallest to largest and compare them to the adjusted threshold, which is calculated by dividing the significance level (0.05) by the number of comparisons (3). If the p-value for a comparison is less than or equal to the adjusted threshold, we reject the null hypothesis for that comparison. Otherwise, we fail to reject the null hypothesis.
To apply the Bonferroni-Holm correction to this experiment, we first need to calculate the mean and standard deviation for each dataset. We can then perform pairwise comparisons using a t-test, assuming equal variance.
The calculations for part a are as follows:
- t-value for comparison between A and B = 3.88
- t-value for comparison between A and C = 5.16
- p-value for comparison between A and B = 0.0035
- p-value for comparison between A and C = 0.0002
Since both p-values are less than 0.05, we reject the null hypothesis and conclude that there is a significant difference between the control group and both experimental groups.
To apply the Bonferroni-Holm correction, we must adjust the significance level for multiple comparisons. In this case, we are making three comparisons (A vs. B, A vs. C, and B vs. C), so we divide the significance level by three: 0.05/3 = 0.0167.
Next, we rank the p-values in ascending order:
1. A vs. B (p = 0.0035)
2. A vs. C (p = 0.0002)
3. B vs. C (p = 0.3)
We compare each p-value to the adjusted threshold:
1. A vs. B (p = 0.0035) is less than or equal to 0.0167, so we reject the null hypothesis.
2. A vs. C (p = 0.0002) is less than or equal to 0.0083, so we reject the null hypothesis.
3. B vs. C (p = 0.3) is greater than 0.005, so we fail to reject the null hypothesis.
Using the Bonferroni-Holm correction, we found that there is a significant difference between the control group (A) and both experimental groups (B and C). However, there is no significant difference between groups B and C. This suggests that both experimental drugs have a similar effect on the physiological variable being measured.
To know more about null hypothesis visit:
https://brainly.com/question/30821298
#SPJ11
simplify (-2) × (6-5) × (-5)
Answers
-2x(6-5)x-5
-2x1x-5
-2x-5
=+10
Answer: 10
Step-by-step explanation:
6-5 = 1
-2 x 1 = -2
-2 x -5 = 10
The sales tax in one town is 8%.So, the total cost of an item can be written as c=0.08cents .what is the total cost of an that sells for $12 ?its due tmrr asap
Answers
Answer: 12.96
Step-by-step explanation: you should multiply 12 * 0.08 = 0.96
then just add 0.96 + 12 = 12.96
Calculate the number of hours needed to frame a one-story house 36' long, in which 2 x 6 x 16 joists will be placed 16" OC (round your answer to nearest whole number. ) A. 4 B. 7 C. 6 D. 5?
Answers
6 Joists are horizontal framing members used to support ceilings or floors, and they're usually made of wood. The correct option is option C. 6.
It should be remembered that the number of joists and their thicknesses must be determined by the intended loading. So, we have to calculate the number of joists needed in order to frame a one-story house 36 feet long, in which 2 x 6 x 16 joists will be placed 16 inches apart in the center. So, we have:
Number of joists required= Total length of house/spacing of joist + 1
= (36×12) / 16 + 1= 28.5 + 1= 29.5 ≈ 30
Therefore, 30 joists are required.
Also, since there are 30 joists and each joist is 16 feet long, the total length of the joists is:
Total length of joists = Length of each joist × Number of joists
= 16 × 30 = 480 feet
Therefore, 480 feet of framing material is required.
To calculate the number of hours required for framing, we can use the following formula:
Time required = (Total length of framing / Length of each piece) × Time required per piece
The time required per piece depends on the type of work, the skill level of the workers, and the equipment being used. Therefore, we can only assume that the time required per piece is 1 hour. So,
Time required = (480 / 16) × 1= 30 × 1= 30
Therefore, 30 hours are required to frame the house. Therefore, the correct option is C. 6.
You can learn more about skill level at: brainly.com/question/31301960
#SPJ11
9m(4m – 3n) – (5m – n)2
A 11m2
– 17 mn – n2
B 11m2
– 37 mn + n2
C 34m2
– 17 mn – n2
D 34m2
– 37 mn + n2
Answers
Answer:
\(36m {}^{2} - 27mn - 10m + 2n\)
Step-by-step explanation:
Use distributive law.
Work with one bracket at a time.
Watch out for the signs!
What is 95 feet in inches
Answers
Answer:
1140 in.
Step-by-step explanation:
Multiply the length value by 12, so 95*12=1140
Have a beautiful day! v(⌒o⌒)v♪
what is the mathematical average of the number of feet in a yard, seconds in a minutes and months in a year? enter a numerical value only.
Answers
The mathematical average of the number of feet in a yard, seconds in a minutes and months in a year is 25.
How to calculate the arithmetic average for the given scenario?In Mathematics and Geometry, the arithmetic average for any data set can be calculated by using the following formula:
Arithmetic average = [∑(x)]/n
Generally speaking, we have the following standard parameters;
There are 3 feet in one (1) yard.There are 60 seconds in one (1) minute.There are 12 months in one (1) year.∑(x) = 3 + 60 + 12
∑(x) = 75
Now, we can determine the arithmetic average as follows;
Arithmetic average = 75/3
Arithmetic average = 25.
Read more on mean here: brainly.com/question/9550536
#SPJ4
Which equation describes the line containing the points (-2,2) and (4, -2)?
A. y = 2x + 6
B. y = -2/3x + 2/3
y = y = -3/2x + 5
y = y = -2/3x + 8/3
Answers
Answer:B
Step-by-step explanation:
find the slope of the line segment joining the pair (7,8) and (-2,3)
Answers
Answer:
\(\frac{5}{9}\)
Step-by-step explanation:
\(\mathrm{Given,}\\\mathrm{(x_1,y_1)=(7,8)}\\\mathrm{(x_2,y_2)=(-2,3)}\\\mathrm{Now,}\\\mathrm{Slope = \frac{y_2-y_1}{x_2-x_1}=\frac{3-8}{-2-7}=\frac{-5}{-9}=\frac{5}{9}}\)
Estimate. Show your work. Cam buys 3 CDs for $26.77. If each CD costs the same amount, about how much does each CD cost?
Answers
Answer:
$8.92
Step-by-step explanation:
26.77/3 ≈ 8.92
Is your answer.
Question 1 (2 x 12 = 24 marks) Analyze and discuss the performance (in Big-O notation) of implementing the following methods over Singly Linked List and Doubly Linked List Data structures: To be submitted through Turnitin.Maximum allowed similaritv is 15% Operation Singly Linked List Doubly Linked List add to start of list Big-O notation Explanation add to end of list Big-O notation Explanation add at given index Big-O notation Explanation
Answers
In analyzing the performance of implementing the given methods over Singly Linked List and Doubly Linked List data structures, we consider the Big-O notation, which provides insight into the time complexity of these operations as the size of the list increases.
Add to Start of List:
Singly Linked List: O(1)
Doubly Linked List: O(1)
Both Singly Linked List and Doubly Linked List offer constant time complexity, O(1), for adding an element to the start of the list.
This is because the operation only involves updating the head pointer (for the Singly Linked List) or the head and previous pointers (for the Doubly Linked List). It does not require traversing the entire list, regardless of its size.
Add to End of List:
Singly Linked List: O(n)
Doubly Linked List: O(1)
Adding an element to the end of a Singly Linked List has a time complexity of O(n), where n is the number of elements in the list. This is because we need to traverse the entire list to reach the end before adding the new element.
In contrast, a Doubly Linked List offers a constant time complexity of O(1) for adding an element to the end.
This is possible because the list maintains a reference to both the tail and the previous node, allowing efficient insertion.
Add at Given Index:
Singly Linked List: O(n)
Doubly Linked List: O(n)
Adding an element at a given index in both Singly Linked List and Doubly Linked List has a time complexity of O(n), where n is the number of elements in the list.
This is because, in both cases, we need to traverse the list to the desired index, which takes linear time.
Additionally, for a Doubly Linked List, we need to update the previous and next pointers of the surrounding nodes to accommodate the new element.
In summary, Singly Linked List has a constant time complexity of O(1) for adding to the start and a linear time complexity of O(n) for adding to the end or at a given index.
On the other hand, Doubly Linked List offers constant time complexity of O(1) for adding to both the start and the end, but still requires linear time complexity of O(n) for adding at a given index due to the need for traversal.
To learn more about Big-O notation visit:
brainly.com/question/30452896
#SPJ11
Which equation can be used to solve for b?
Triangle A B C is shown. Angle B C A is a right angle and angle C A B is 30 degrees. The length of side B C is 5 centimeters, the length of B A is 10 centimeters, and the length of C A is b.
tan(30o) = StartFraction 5 Over b EndFraction
tan(30o) = StartFraction b Over 5 EndFraction
tan(30o) = StartFraction 10 Over b EndFraction
tan(30o) =
Answers
The equation can be used to solve for b is b = 5 / tan30°
What is an equation?An equation is a mathematical statement that shows that two mathematical expressions are equal.
Given that, a triangle, ∠ BCA = 90°, ∠ CAB = 30°, we need to find the equation for AC,
The equation for AC(b) =
tan30° = 5/b
b = 5 / tan30°
Hence, the equation can be used to solve for b is b = 5 / tan30°
Learn more about equation, click;
https://brainly.com/question/29657992
#SPJ1
A business advertises that everything in the store is an additional 10% off the already reduced prices. marcus picks out 2 shirts that are on a 30% off rack. if the shirts are originally priced at $28.99 and $30.29 and there is 6% sales tax, how much does marcus end up paying for them? a. $39.59 b. $37.70 c. $37.35 d. $35.57
Answers
The amount that marcus end up paying for the two shirts originally priced at $28.99 and $30.29 is given by: Option A: $39.59
How to find the percentage from the total value?Suppose the value of which a thing is expressed in percentage is "a'
Suppose the percent that considered thing is of "a" is b%
Then since percent shows per 100 (since cent means 100), thus we will first divide the whole part in 100 parts and then we multiply it with b so that we collect b items per 100 items(that is exactly what b per cent means).
Thus, that thing in number is
\(\dfrac{a}{100} \times b\)
For this case, we're given that:
Cost of two shirts originally is $28.99 and $30.29
30% is off on each of them, so we get:
New price of each shirt = Old price of each shirt - 30% of old price of each shirt
New price of each shirt = \(28.99 - \dfrac{28.99}{100} \times 30\) and \(30.29- \dfrac{30.29}{100} \times 30\)
This sums up to $41.496
Now, at this price, there'd be application of 10% additional discount.
That will make the new price as:
\(41.496 - \dfrac{41.496}{100} \times 10 =37.3464\)
Now, on this price, there'd be applied 6% sales tax
So, the amount he paid for 2 shirts = discounted price + 6% tax on discounted amount
= \(37.3464 + \dfrac{37.3464}{100} \times 6 \approx 39.59 \: \rm dollars\)
Thus, the amount that marcus end up paying for the two shirts originally priced at $28.99 and $30.29 is given by: Option A: $39.59
Learn more about percent here:
https://brainly.com/question/11549320
Find the range of the data set 7,510,9,15,12,6,9,7,10
I NEED HELP WHIT ALL MY WROK TODAY GUYS SO ANY ONE HELP ONLY WHIT MATH THEY ARE GRADE TODAYYY
Answers
Answer: 10
Step-by-step explanation:
So to find range you take the highest number and subtract it by the lowest
Answer:highest take lowest,15-5=10
The range is 10.
Step-by-step explanation:
1. Differentiate the function f(x) = ln (81 sin^2 (x)) f’(x) 2. Differentiate the function P(t) = in ( √t2 + 9) p' (t) 3. if x2 + y2 + z2 = 9, dx/dt = B, and dy/dt = 4, find dz/dt when (x,y,z) = (2,2,1)
dz/dt =
Answers
First you will get 4dz
Solve by factorising.
x2 + 10x + 21 = 0
Answers
Answer:
×=-7,×=-3
Step-by-step explanation:
plz don't mind my "×"
×^2+10×+21=0
×^2+7×+3×+21=0
rearrange
that is ; ×^2+3×+7×+21=0
(×^2+3×)+(7×+21)=0
×(×+3)+7 (×+3)=0
therefore
(×+7) (×+3)=0
×+7=0 ×+3=0
×= -7 ×=-3
Which statement is true about the ranges for the box plots? the range of the morning box plot is the same as the range of the afternoon box plot. The range of the morning box plot is 1 less than the range of the afternoon box plot. The range of the morning box plot is 1 more than the range of the afternoon box plot. The range of the morning box plot is 2 less than the range of the afternoon box plot.
Answers
The range of the Morning box plot is the same as the range of the Afternoon box plot. Therefore, the correct answer is option A.
A box and whisker plot—also called a box plot—displays the five-number summary of a set of data. The five-number summary is the minimum, first quartile, median, third quartile, and maximum. In a box plot, we draw a box from the first quartile to the third quartile. A vertical line goes through the box at the median.
Number of sales in Afternoon:
Minimum value = 4
First quartile = 8
Median = 14
Third quartile = 15
Maximum value = 16
Here, the range is 16-4=12
Number of sales in Morning:
Minimum value = 3
First quartile = 5
Median = 8
Third quartile = 12
Maximum value = 15
Here, the range is 15-3=12
Therefore, the correct answer is option A.
Learn more about the box plot here:
https://brainly.com/question/1523909.
#SPJ6
Which point is located at (2, 3.5)?
a
b
c
d
Answers
Answer: C
Step-by-step explanation:
Answer:
its c
Step-by-step explanation:
hope it helps
fastt
13. Calculate the compound interest of an annuity due of BD400 paid each 4 months for 6.2 years if the nominal rate is 3% thirdly? (3 Points)
Answers
Therefore, the compound interest of the annuity due of BD 400 paid each 4 months for 6.2 years at a nominal rate of 3% per annum is BD 40,652.17.
Compound interest of an annuity due can be calculated using the formula:A = R * [(1 + i)ⁿ - 1] / i * (1 + i)
whereA = future value of the annuity dueR = regular paymenti = interest raten = number of payments First, we need to calculate the effective rate of interest per period since the nominal rate is given per annum. The effective rate of interest per period is calculated as
:(1 + i/n)^n - 1 = 3/1003/100 = (1 + i/4)^4 - 1
(1 + i/4)^4 = 1.0075i/4 = (1.0075)^(1/4) - 1i = 0.0303So,
the effective rate of interest per 4 months is 3.03%.Next, we can substitute the given values in the formula:
A = BD 400 * [(1 + 0.0303)^(6.2 * 3) - 1] / 0.0303 * (1 + 0.0303)A = BD 400 * [4.227 - 1] / 0.0303 * 1.0303A = BD 400 * 101.63A = BD 40,652.17
For such more question on interest
https://brainly.com/question/25720319
#SPJ8
Which could you use to show that u || v?
PLZ ANSWER ASAP!!!!!
Answers
Answer:
3 and 7
Step-by-step explanation:
Suppose that π(x) = F(x) for some strictly increasing cdf F. Explain why a mono- tone transformation of x exists such that the logistic regression model holds. Gener- alize to alternative link functions.
Answers
A monotone transformation of x exists such that the logistic regression model holds because the relationship between the predictor variable x and the probability of success π(x)
Success π(x) can be transformed using a monotone function to achieve linearity, which is a fundamental assumption of logistic regression.
In logistic regression, the goal is to model the relationship between a predictor variable x and the probability of success π(x). The logistic regression model assumes that the relationship between x and π(x) can be represented by a sigmoidal curve.
However, if the relationship between x and π(x) does not exhibit linearity, a monotone transformation of x can be applied to achieve linearity.
If we have π(x) = F(x),
where F(x) is a strictly increasing cumulative distribution function (cdf), we can find a monotone transformation g(x) such that g(F(x)) results in a linear relationship.
The transformed predictor variable g(x) can be used in the logistic regression model to achieve linearity.
This concept can be generalized to alternative link functions in logistic regression. Depending on the specific relationship between the predictor variable x and the probability of success π(x), different link functions (e.g., logit, probit, complementary log-log) can be chosen to model the relationship and achieve linearity through appropriate monotone transformations if necessary.
The choice of the link function depends on the nature of the data and the assumptions made about the relationship between x and π(x).
To know more about monotone transformation, refer here:
https://brainly.com/question/14761350#
#SPJ11
What is 6.33 time as 6
Answers
Answer: 37.98 (I assume you mean times)
Step-by-step explanation:
Calculator :)
A relation that assigns to each element x from a set of inputs, or __________ , exactly one element y in a set of outputs, or ___________ , is called a
Answers
A relation that assigns to each element x from a set of inputs, or domain, exactly one element y in a set of outputs, or range, is called a function.
This is further explained below.
What is a function?Generally, A mathematical function from set X to set Y gives each element of X a unique value in Y. The set X is known as the function's domain and the set Y is its codomain. The original function was a simplification of the relationship between two variables.
In conclusion, A relation is said to be a function if it assigns precisely one element y from a set of outputs to each and every one of the domain's inputs.
Read more about the function
https://brainly.com/question/12431044
#SPJ1
if a card is picked at random from a standard 52-card deck, what is the probability of getting a diamond or a king? give your answer as a fraction.
Answers
The probability of getting a diamond or a king from a standard 52-card deck is 4/13.
In mathematics, probability is a measure of the likelihood that an event will occur. It is a way to quantify the chances of different outcomes or events in a random experiment or process. Probability is typically expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event
In a standard deck of 52 cards, there are 13 diamonds and 4 kings. However, the king of diamonds is also a diamond, so we must subtract one from the total count to avoid double-counting.
So there are 13 + 4 - 1 = 16 cards that are either diamonds or kings.
The probability of picking a diamond or a king is therefore 16/52.
Simplifying the fraction, we get:
16/52 = 4/13
Learn more about probability here
brainly.com/question/11234923
#SPJ4
Which of the following is the maximum value of the function y = -x2 + 2x + 1?
Answers
We first complete the square.
y=-x^2+2x+1
y=-x^2+2x-1+1+1
y=-(x-1)^2+2
The maximum point is (1,2) since -1 in ( ) meant 1 unit to the right and +2 meant 2 units up.
The maximum point is (1,2) but the maximum value means the y value which is simply just 2.
Done!
Perform the operation and write the result in standard form. (6-3i)(3-3i)
Answers
The expression (6-3i)(3-3i) when simplified is 9 - 27i
How to evaluate the expressionFrom the question, we have the following parameters that can be used in our computation:
(6-3i)(3-3i)
Open the brackets, so we have
(6-3i)(3-3i) = 18 - 9i - 18i - 9
Evaluate the like terms
(6-3i)(3-3i) = 9 - 27i
Hence, the simplified expression is 9 - 27i
Read more about complex numbers at
https://brainly.com/question/8847670
#SPJ1
when a satellite reads radiation from a mountain the amount of radiation it observes is distributed n(490, 2916) (units are msv). a spy satellite has detected a radiation level of 599 from a mountain known to have terrorists. assuming there is no nuclear danger here, what is the probability of a random radiation measurement being 599 or higher?
Answers
The probability of a radiation measurement of 599 or higher from a mountain known to have terrorists, assuming no nuclear danger, is about 0.0668.
How to find the probability?We are given that the radiation levels observed by the satellite are normally distributed with a mean of 490 and a variance of 2916. We want to find the probability of a random radiation measurement being 599 or higher, assuming there is no nuclear danger.
First, we need to standardize the radiation level of 599 using the formula:
z = (x - mu) / sigma
where x is the radiation level, mu is the mean, and sigma is the standard deviation. Substituting the values we have:
z = (599 - 490) / √(2916) = 1.5
Now, we can use a standard normal distribution table or calculator to find the probability of a z-score of 1.5 or higher. The table or calculator will give us the area under the standard normal curve to the right of 1.5.
Using a calculator, we can find this probability as follows:
P(Z > 1.5) = 0.0668 (rounded to four decimal places)
Therefore, the probability of a random radiation measurement being 599 or higher is approximately 0.0668, assuming there is no nuclear danger.
Learn more about Probability
brainly.com/question/16484393
#SPJ11
Find f(1/2) for the function:
f(x)=-x^2+x-3
Answers
Answer:
this i think
Step-by-step explanation:
Solve the following question
Answers
Answer:
x=-1
Step-by-step explanation:
First we can square both sides to remove the square roots:
\((\sqrt{x+7})^{2} +(\sqrt{x+2})^{2} = (\sqrt{6x+13})^{2}\)
Simplify:
\(x+7+x+2=6x+13\)
Now we simplify further:
\(2x+9=6x+13\)
Now we subtract 13 from both sides:
\(2x+9-13=6x+13-13\)
Simplify:
\(2x-4=6x\)
Subtract 2x from both sides:
\(2x-4-2x=6x-2x\)
Simplify:
\(-4=4x\)
Divide both sides by 4:
\(\frac{-4}{4} = \frac{4x}{4}\)
Simplify:
-1=x
x=-1
find a formula for the probability of the union of five events in a sample space if no four of them can occur at the same time.
Answers
The formula for the probability is as follows:
P(A ∪ B ∪ C ∪ D ∪ E) = P(A) + P(B) + P(C) + P(D) + P(E) - P(A ∩ B) - P(A ∩ C) - P(A ∩ D) - P(A ∩ E) - P(B ∩ C) - P(B ∩ D) - P(B ∩ E) - P(C ∩ D) - P(C ∩ E) - P(D ∩ E) + P(A ∩ B ∩ C) + P(A ∩ B ∩ D) + P(A ∩ B ∩ E) + P(A ∩ C ∩ D) + P(A ∩ C ∩ E) + P(A ∩ D ∩ E) + P(B ∩ C ∩ D) + P(B ∩ C ∩ E) + P(B ∩ D ∩ E) + P(C ∩ D ∩ E) - P(A ∩ B ∩ C ∩ D) - P(A ∩ B ∩ C ∩ E) - P(A ∩ B ∩ D ∩ E) - P(A ∩ C ∩ D ∩ E) - P(B ∩ C ∩ D ∩ E) + P(A ∩ B ∩ C ∩ D ∩ E).
To calculate the probability of the union of five events in a sample space, we use the principle of inclusion-exclusion. The formula takes into account all possible combinations of the events and adjusts for overlaps.
The formula starts with adding the individual probabilities of each event: P(A) + P(B) + P(C) + P(D) + P(E). This accounts for the events occurring individually.
Then, we subtract the probabilities of the intersections of two events: P(A ∩ B), P(A ∩ C), P(A ∩ D), P(A ∩ E), P(B ∩ C), P(B ∩ D), P(B ∩ E), P(C ∩ D), P(C ∩ E), P(D ∩ E). This ensures that the overlapping probabilities are not double-counted.
Next, we add back the probabilities of the intersections of three events: P(A ∩ B ∩ C), P(A ∩ B ∩ D), P(A ∩ B ∩ E), P(A ∩ C ∩ D), P(A ∩ C ∩ E), P(A ∩ D ∩ E), P(B ∩ C ∩ D), P(B ∩ C ∩ E), P(B ∩ D ∩ E), P(C ∩ D ∩ E). This compensates for the previously subtracted probabilities.
We continue this pattern of subtraction and addition for the intersections of four events and five events.
Finally, we subtract the probability of the intersection of all five events: P(A ∩ B ∩ C ∩ D ∩ E). This ensures that it is not counted multiple times during the inclusion-exclusion process.
By following this formula, we can calculate the probability of the union of five events in a sample space, satisfying the condition that no four of them can occur simultaneously.
Learn more about Probability:
brainly.com/question/32117953
#SPJ11