,P(X ≥ 10) = 1 - P(X < 10) = 1 - 0.0000380 = 0.9999620 (rounded to 7 decimal places)The probability that 10 or more of the flights were on time is 0.9999620, or approximately 1.0.
To find the probability that 10 or more of the flights were on time, we need to use the binomial distribution formula, which is given as:P(X = k) = nCk * p^k * (1-p)^(n-k)Where P(X = k) is the probability of k successes, n is the total number of trials, p is the probability of success on a single trial, and nCk is the number of combinations of n things taken k at a time.To apply this formula to the given problem, we need to identify the values of n, k, and p. From the problem statement, we know that there were a total of 60 flights, and we want to find the probability of 10 or more of them being on time. Therefore, n = 60 and k ≥ 10. The probability of a single flight being on time is not given, so we cannot use it directly. However, we can use the fact that the overall percentage of flights that were on time is 80%, or 0.8. This means that p = 0.8.To find the probability that 10 or more of the flights were on time, we need to add up the probabilities of all the possible values of k that meet this criterion. That is:P(X ≥ 10) = P(X = 10) + P(X = 11) + ... + P(X = 60)nC10 * p^10 * (1-p)^(n-10) + nC11 * p^11 * (1-p)^(n-11) + ... + nC60 * p^60 * (1-p)^(n-60)Using a calculator or computer software, we can calculate each of these probabilities and then add them up. However, this would be quite time-consuming. Instead, we can use the complement rule to find the probability that fewer than 10 of the flights were on time, and then subtract this from 1. That is:P(X ≥ 10) = 1 - P(X < 10)P(X < 10) = P(X = 0) + P(X = 1) + ... + P(X = 9)nC0 * p^0 * (1-p)^(n-0) + nC1 * p^1 * (1-p)^(n-1) + ... + nC9 * p^9 * (1-p)^(n-9)Again, we can use a calculator or software to find each of these probabilities and add them up. Doing so gives:P(X < 10) = 0.0000380 (rounded to 7 decimal places)
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The probability that 10 or more flights were on time is approximately 0.9992 or 99.92%.
To find the probability that 10 or more of the flights were on time, we need to use the binomial distribution formula which is given by;
P(X = k) =[tex](nCk) * p^k * (1 - p)^(n - k)[/tex]
Where;n is the total number of flights, and p is the probability of a flight being on time.
k is the number of flights that are on time.
We are given;
n = 15 flights
p = 0.70
The probability that a flight will be on time k ≥ 10, that is 10 or more flights are on time.
Now we can solve for the probability as follows;
P(X ≥ 10) = P(X = 10) + P(X = 11) + ... + P(X = 15)
P(X ≥ 10) = [tex](15C10 * 0.70^10 * 0.30^5) + (15C11 * 0.70^11 * 0.30^4) + (15C12 * 0.70^12 * 0.30^3) + (15C13 * 0.70^13 * 0.30^2) + (15C14 * 0.70^14 * 0.30^1) + (15C15 * 0.70^15 * 0.30^0)[/tex]
Using a calculator, we get;
P(X ≥ 10) = 0.9992
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What is the value of x?
5(x+2)=11
Answer:
X=1.8
Step-by-step explanation:
Step one: 5 times 1.8 equals 9.
Step two: 9+2=11
What value for the variable makes this equation true?
`-2= 3+ b/4
<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3
Answer:
Let's solve your equation step-by-step.
−2 = 3 + b / 4
Step 1: Simplify both sides of the equation.
−2 = 3 + b / 4
−2 = 3 + 1 / 4b
−2 = 1 / 4b + 3
Step 2: Flip the equation.
1 / 4b + 3 = −2
Step 3: Subtract 3 from both sides.
1 / 4b + 3 − 3 = −2 − 3
1 / 4b = −5
Step 4: Multiply both sides by 4.
4 * ( 1 / 4b ) = ( 4 ) * ( −5 )
b = −20
Answer:
b = −20
<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3
please help i’m desperate
Answer:
4.52
Step-by-step explanation:
volume of cone is
[tex]\pi {r}^{2} \frac{h}{3} [/tex]
h is 2.2
and
r is 1.4
When a coin is flipped 20 times and lands heads up 11 times, what is the
experimental probability? Write your answer as a decimal.
Answer:
0.55
Step-by-step explanation:
11 (the time it lands on heads) /20 (the number of times it was flipped) = 0.55
HELP PLEASE!! (No links)
A plane slices horizontally through a cone as shown, which term best describes the cross-section?
es )
A)
circle
B)
rectangle
C)
rhombus
D)
triangle
Select the correct answer.
If you divide-16 into 5 equal parts, how much is each part equal to?
A.
-4
B.
-3.2.
C.
+2.4
D.
+5
Reset
Next
Answer:
-3.2
Step-by-step explanation:
If the vibrating string is subject to viscous damping, the governing equa- tion can be written as du(x, t) J²u(x, t) Ət² ²u(x, t) x2 2h = 7 Ət where h is a constant.
The given equation represents the governing equation for a vibrating string subject to viscous damping. The equation involves the partial derivatives of the displacement function u(x, t) with respect to both time t and position x. The left-hand side of the equation captures the effects of inertia and elasticity of the string, while the right-hand side represents the damping term.
The term du(x, t)/dt^2 describes the acceleration of the string at a specific point (x, t) and is proportional to the second derivative of u with respect to time. The term d^2u(x, t)/dx^2 represents the curvature or bending of the string at that point and is proportion to the second derivative of u with respect to position.
The constant h represents the damping coefficient, which determines the level of damping present in the system. Higher values of h indicate stronger damping effects, while lower values imply less damping.
The equation states that the sum of the damping term and the terms related to inertia and elasticity is equal to 7 times the partial derivative of u with respect to time.
By solving this governing equation, one can determine the behavior of the vibrating string over time and space, taking into account the effects of viscous damping.
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2x+4 and 5x-8 help please
Answer:
7x - 4
Step-by-step explanation:
2x + 4 + 5x - 8
7x + 4 - 8
7x - 4
Suppose that V₁, V2, , Um are linear dependent in a vector space V. For every & EV, show that 7₁, V2, , Um, 7 are also linearly dependent. 9
By supposing that V₁, V2, and Um are linearly dependent in a vector space V. For every & EV. V₁, V₂, ..., Uₘ are linearly dependent in a vector space V. As 7₁V₁ = 7(c₂V₂ + c₃V₃ + ... + cₘUₘ)
To show that 7₁V₁, 7₂V₂, ..., 7ₘUₘ, 7 is also linearly dependent for every 'c' in V, we can use the following approach: If V₁, V₂, ..., Uₘ are linearly dependent, then we can write at least one vector as a linear combination of the other vectors.
Let's assume V₁ can be written as a linear combination of the other vectors as follows:
V₁ = a₂V₂ + a₃V₃ + ... + aₘUₘ
where a₂, a₃, ..., aₘ are constants. Now, we can express the vector 7₁V₁ as:
7₁V₁ = 7₁a₂V₂ + 7₁a₃V₃ + ... + 7₁aₘUₘ
Since 7 is a constant, we can take it outside the bracket as:
7₁V₁ = 7(a₂7₁V₂ + a₃7₁V₃ + ... + aₘ7₁Uₘ)
Let's assume the sum inside the bracket is equal to 'b'. Then,
7₁V₁ = 7b
Since we know that V₁, V₂, ..., Uₘ are linearly dependent, we can write b as a linear combination of the other vectors as follows:
b = c₂V₂ + c₃V₃ + ... + cₘUₘ
where c₂, c₃, ..., cₘ are constants. Now, substituting the value of b in the equation for 7₁V₁, we get:
7₁V₁ = 7(c₂V₂ + c₃V₃ + ... + cₘUₘ)
This shows that 7₁V₁, 7₂V₂, ..., 7ₘUₘ, 7 are also linearly dependent. Therefore, we have proved that for every 'c' in V, 7₁V₁, 7₂V₂, ..., 7ₘUₘ, 7 are also linearly dependent.
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The center and a point on a circle are given. Find the circumference to the nearest tenth.
center:(5, −5);
point on the circle: (25, 10)
The circumference is about ?
Answer: 50pi or about 157
Step-by-step explanation: Find the distance between the two points using the Pythagorean theorem, and since that is the radius double it to get the diameter, then multiply that by pi to get the answer. Hope this helps!
Answer:
Circumference ~ 157
Step-by-step explanation:
Use the distance formula to find the radius. The radius is 25. Plug 25 into the formula 2(pi)r to get 50pi or about 157.
You draw one card from a deck. If you do
this 100 times, how many times would you
expect that it's a red 3?
Around 4 times, maybe 3
Step-by-step explanation:
there's 2 red 3s and there's 52 cards in a deck. so u have a 2/52 chance to get a red 3 so its about a 4/104 chance to get a red 3 if u draw 1 card 100 times
Can pls someone help I need help pls
(27.26604445073)
this answer for this question
specify a codomain for each of these functions in exercise 16. under what conditions is each of these funtions with the codomain you specified onto?
The codomain of a function is the set that contains all possible values that the function can map to. It represents the range of possible output values. To specify a codomain for a function, you need to consider the nature of the function and the type of values it can produce.
A function is considered onto (or surjective) if every element in the codomain has at least one corresponding element in the domain that maps to it. In other words, for each value in the codomain, there exists an input in the domain that produces that particular output.
To determine if a function is onto, you need to ensure that every element in the codomain is reached by the function. This can be achieved by satisfying certain conditions, such as:
The range of the function (the actual set of output values) is equal to the codomain. This means that the function covers all possible values in the codomain.
The function is defined for every element in the codomain. There are no "gaps" or missing elements that the function does not cover.
The function is one-to-one (injective). This means that each element in the domain maps to a unique element in the codomain, preventing any overlap or repetition.
These conditions ensure that every value in the codomain is covered by the function, making it onto.
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Please help me solve this.
Answer: 59
Step-by-step explanation:
Given
2b³+5 and b=3 ⇒ 2b³+5=2(3)³+5
Simplify exponents
2(3)³+5
=2(3×3×3)+5
=2×27+5
Multiplication
=54+5
Addition
=59
Hope this helps!! :)
Please let me know if you have any questions
If f (x) = 3x - 2 and fog(x) = 6x - 2 then find the value of
x such that gof(x) = 8.
9514 1404 393
Answer:
x = 2
Step-by-step explanation:
To find g(x), we can start with the inverse of f(x).
f(y) = x . . . . . . . solve this to find f^-1(x)
3y -2 = x
3y = x +2 . . . . add 2
y = (x +2)/3 = f^-1(x) . . . . divide by 3
__
Now, we can find g(x):
f^-1(f(g(x)) = g(x)
f^-1(6x -2) = ((6x -2) +2)/3 = g(x)
6x/3 = g(x) = 2x
__
Now, we want g(f(x)) = 8
g(3x -2) = 8
2(3x -2) = 8
6x -4 = 8
6x = 12
x = 2 . . . . makes g(f(x)) = 8
I've done everything I can, I cant figure this out pls help asap
1. H
2. A
3. E
4. C
I have to type extra words to turn this in so basically I just found the slop of the line then matched it with the letters...
Help ME????????????????????
Answer:
3 x -10.5
Step-by-step explanation:
The problem says "each for -10.50." That means every entry has a change of -10.5.
We know there are 3 entries so we are going to multiply 3 x -10.5 (The expression is the answer, not the actual answer.)
a) Find the general solution y=yc+yp of the differential equation
y'' + x^2 y' +2xy = 5-2x+10x^3
that consists of three power series centered at x =0. You can list the first five nonzero terms of each power
series.
b) Consider the initial value problem
y' = √1-y^2 y(0)=0
Show that y= sin x is the solution of the initial value problem (b).
c) Look for a solution of the initial value problem (b) in the form of a power series about x = 0. Find
the coefficients up to the term in x^7 in this series.
a) To find the general solution of the given differential equation, a power series centered at x=0 is used, and the first five nonzero terms of each power series are determined.
b) The solution to the initial value problem y' = √(1-y^2), y(0) = 0, is shown to be y = sin(x).
c) The coefficients up to the term in x^7 are found for a power series solution of the initial value problem y' = √(1-y^2), y(0) = 0.
a) To find the general solution y = yc + yp of the given differential equation:
y'' + x^2 y' + 2xy = 5 - 2x + 10x^3,
we can first find the complementary solution yc by assuming a power series of the form y = ∑(n=0 to ∞) a_n x^n. Substituting this series into the differential equation and equating coefficients of like powers of x, we can determine the values of the coefficients a_n. However, for simplicity, we will only consider the first five nonzero terms of the power series.
Let's write the power series for yc:
yc = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + ...
Differentiating twice with respect to x, we get:
y' = a_1 + 2a_2 x + 3a_3 x^2 + 4a_4 x^3 + ...
y'' = 2a_2 + 6a_3 x + 12a_4 x^2 + ...
Substituting these series into the differential equation, we have:
(2a_2 + 6a_3 x + 12a_4 x^2 + ...) + x^2(a_1 + 2a_2 x + 3a_3 x^2 + 4a_4 x^3 + ...) + 2x(a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + ...) = 5 - 2x + 10x^3
To equate coefficients, we match the powers of x on both sides of the equation:
For the term without x:
2a_2 + a_0 = 5
For the term with x:
6a_3 + 2a_2 + a_1 = -2
For the term with x^2:
12a_4 + 3a_3 + 2a_1 + a_2 = 0
For the term with x^3:
4a_4 + 4a_2 + a_3 = 10
For the term with x^4:
a_4 = 0 (no coefficient on the right-hand side)
Solving this system of equations will give us the values of a_0, a_1, a_2, a_3, and a_4. Since we are only interested in the first five nonzero terms of the power series, we will truncate the series at the fifth term.
b) To show that y = sin(x) is the solution to the initial value problem y' = √(1-y^2), y(0) = 0:
We can differentiate y = sin(x) to obtain y' = cos(x). Substituting this into the differential equation, we have:
cos(x) = √(1 - sin^2(x))
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we simplify the equation to:
cos(x) = √(cos^2(x))
Taking the positive square root, we have:
cos(x) = cos(x)
This confirms that y = sin(x) satisfies the differential equation y' = √(1-y^2).
c) To find a power series solution for the initial value problem y' = √(1-y^2), y(0) = 0, we assume a power series of the form y = ∑(n=0 to ∞) a_n x^n. Substituting this series into the differential equation and equating coefficients, we can determine the values of the coefficients a_n up to the term in x^7.
Let's write the power series for y:
y = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5 + a_6 x^6 + a_7 x^7 + ...
Differentiating y with respect to x, we get:
y' = a_1 + 2a_2 x + 3a_3 x^2 + 4a_4 x^3 + 5a_5 x^4 + 6a_6 x^5 + 7a_7 x^6 + ...
Substituting these series into the differential equation, we have:
a_1 + 2a_2 x + 3a_3 x^2 + 4a_4 x^3 + 5a_5 x^4 + 6a_6 x^5 + 7a_7 x^6 + ... = √(1 - (a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5 + a_6 x^6 + a_7 x^7 + ...)^2)
Simplifying this equation and equating coefficients of like powers of x, we can determine the values of the coefficients a_n up to the term in x^7.
To find the coefficients up to the term in x^7, you will need to perform the substitution and equate coefficients. It will involve expanding the square root and equating coefficients of each power of x from 0 to 7 on both sides of the equation.
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The volume of a triangular pyramid is 13.5 cubic meters. What is the volume of a triangular prism with a congruent base and the same height?
Answer:
The prism is 3 by 3 by 1.5.
Step-by-step explanation:
The prism has a length x and a width x. Since it is a square at the base both length and width are the same amount x. The height is "half the length of one edge of the base". Since the base is x, this makes it 1/2x.
The volume's prism is found using V = l*w*h. Substitute and simplify.
13.5 = x*x*1/2x
13.5 = 1/2 x^3
27 = x^3
3 = x
The prism is 3 by 3 by 1.5.
brainliest?
What is 4x+6<22 ? I fleet getting incorrect answers
Answer:
The answer is x < 4
Step-by-step explanation:
1) Subtract 6 from both sides.
[tex]4x < 22 - 6[/tex]
2) Simplify 22 - 6 to 16.
[tex]4x < 16[/tex]
3) Divide both sides by 4.
[tex]x < \frac{16}{4} [/tex]
4) Simplify 16/4 to 4.
[tex]x < 4[/tex]
Therefor, the answer is x < 4.
is 1563/25 a rational number
Answer:
Yes, it's a rational number
Step-by-step explanation:
can be written as fraction and integers
A researcher wanted to investigate the relationship between gender and career choice (career choice is classified as either predominantly male, predominantly female, or neutral). One-hundred males and one-hundred females were randomly selected and their profession was classified into one of the three career types. Which of the following procedures is most appropriate for the above example?
Chi-square test of independence
One-way within subjects ANOVA (Repeated measures ANOVA)
One-way between subjects ANOVA
Correlation
Chi-square goodness of fit test
The most appropriate procedure for the given example is the Chi-square test of independence.
Chi-square test of independence: This statistical test is used to determine if there is a significant association between two categorical variables. In this case, the researcher wants to investigate the relationship between gender (male or female) and career choice (predominantly male, predominantly female, or neutral). The test will help determine if there is a dependency between gender and career choice.
a. Formulate hypotheses: The null hypothesis (H0) states that there is no association between gender and career choice, while the alternative hypothesis (Ha) states that there is an association.
b. Set the significance level (alpha): Typically, it is set to 0.05 or 0.01, depending on the desired level of confidence.
c. Create a contingency table: Construct a table that shows the observed frequencies of career choices for each gender.
d. Calculate expected frequencies: Compute the expected frequencies under the assumption of independence between gender and career choice.
e. Calculate the chi-square statistic: Determine the chi-square statistic based on the observed and expected frequencies.
f. Determine the critical value: Look up the critical value from the chi-square distribution table using the degrees of freedom and chosen alpha level.
g. Compare the chi-square statistic and critical value: If the chi-square statistic is greater than the critical value, reject the null hypothesis and conclude that there is a significant association between gender and career choice. If it is not, fail to reject the null hypothesis.
h. Interpret the results: Based on the conclusion, provide an interpretation of the findings, indicating whether there is evidence of a relationship between gender and career choice.
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Help please I’m confused
Answer:
it is the 2nd option
Step-by-step explanation:
Please answer now ! 90 points
Step-by-step explanation:
2+3b[tex]\leq[/tex]25b=books
3b[tex]\leq[/tex]23
[tex]b\leq 23/3[/tex]
[tex]b\leq 7.67[/tex]
50+25L[tex]\leq[/tex]200L=lesson
25L[tex]\leq[/tex]150
L[tex]\leq[/tex]6
Hope that helps :)
Answer: a)7, B)8
Step-by-step explanation:
Problem a) If Kai wants to buy just one poster that costs 2 dollars, he has 25-2=23 dollars left. If each book is 3 dollars, then he can buy 23/3 books. The inequality that results is that if b=#ofbooks, then b< or = (25-2)/3, because you cant buy a book for 2 and a half dollars, hence the less than. 23/3 is 7 r2. We get that b< or = to 7 2/3. However, he can't buy 2/3 of a book, so 7. The final inequality we get is that 2+3b<=25
Problem 2) She spends 50 initially, so 200-50=150 dollars left. Thus the number of lessons, or n, cover the rest of the money. Using the same thory as above, the final inequality is 50+25n<=200. n=8.
is the work shown below correct? explain your answer. (11 2i ) – (3 – 10i ) = 11 2i – 3 – 10i = (11 – 3) (2i – 10i) = 8 – 8i
The work shown is not correct. The correct result of (11 + 2i) - (3 - 10i) is 8 + 12i.
To evaluate the given expression, we need to subtract the real parts and the imaginary parts separately.
Real part:
11 - 3 = 8
Imaginary part:
2i - (-10i) = 2i + 10i = 12i
Combining the real and imaginary parts, we get the correct result:
8 + 12i
So, the correct answer is 8 + 12i, not 8 - 8i as shown in the work.
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Part 2: Each number is worth 3 points. Partial credit may be given. 3. Mrs. Reyes wrote 8 tenths minus 2 hundredths on the board. Sammy said the answer is 6 tenths because 8 minus 2 is 6. Is he correct? Explain.
No, Sammy's answer of 6 tenths is not correct because the answer is 7 tenths.
Is Sammy correct in his answer?To know if Sammy's answer is correct, we will perform subtraction:
8 tenths - 2 hundredths
In this case, we wil convert 8 tenths to hundredths by multiplying it by 10:
8 tenths = 8 * 10
8 tenths = 80 hundredths
==> 80 hundredths - 2 hundredths
==> 78 hundredths
Converting 78 hundredths back to tenths, we divide by 10:
==> 78 hundredths / 10
==> 7.8 tenths.
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A social researcher wants to test the hypothesis that college students who drink alcohol while text messaging type a different number of keystrokes than those who do not drink while they text. The social researcher studies a sample 50 drinking texters and 50 non-drinking texters. The sample mean number of keystrokes for drinking texters was 142 with a sample standard deviation of 7.45. The sample mean number of keystrokes for non-drinking texters was 120 with a sample standard deviation 6.81. Test the null hypothesis of no difference in mean keystrokes between the population of students who drink while texting and the population of students who do not drink while texting.
What is the null hypothesis in this study?
There is no difference in mean keystrokes between the populations of students who drink and text and students who do not drink and text.
There is a difference in mean keystrokes between the populations of students who drink and text and students who do not drink and text.
The sample mean keystrokes for drinking texters is ______ and the sample mean keystrokes for non-drinking texters is _______
The sample variance for drinking texters is _____ and the sample variance for non-drinking texters is ________
What is the standard error of the difference between means?
What is the calculated t? _____
What is the critical t? ______
Based on the comparison of calculated t and critical t, what must we do?
Retain the null hypothesis of no difference in mean keystrokes between the populations of drinking texters and non-drinking texters.
Reject the null hypothesis and conclude there is a difference in mean keystrokes between the populations of drinking texters and non-drinking texters.
The null hypothesis in this study is that there is no difference in mean keystrokes between the populations of students who drink and text and those who do not.
The null hypothesis in this study states that there is no difference in mean keystrokes between the populations of students who drink and text and those who do not. It assumes that the mean keystrokes for both groups are equal.
The sample means keystrokes for drinking texters is stated as 142, while for non-drinking texters, it is 120.
The sample variances for drinking and non-drinking texters are not provided in the question. Without the sample variances, it is not possible to calculate the standard error of the difference between means.
The calculated t-value and critical t-value are not given in the question. The comparison between the calculated t-value and critical t-value is necessary to determine the appropriate action.
Based on the comparison of the calculated t-value and critical t-value, the appropriate action to take is not specified in the question. It is usually done by comparing the calculated t-value with the critical t-value at a specific significance level. If the calculated t-value exceeds the critical t-value, the null hypothesis is rejected, suggesting a significant difference in mean keystrokes between the populations. If the calculated t-value is less than the critical t-value, the null hypothesis is retained, indicating no significant difference in mean keystrokes between the populations.
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"You go shopping and notice that 25 kg of PEI’s Famous Potatoes cost $12.95, and 10 kg of Idaho’s Potatoes cost $5.78.
Which is the better deal?
Justify your answer."
Answer:
PEI famous potato is a better deal
Step-by-step explanation:
Given that :
PEI potato :
25kg costs $12.95
Price per kg :
$12.95 / 25 = $0.518 per kg
IDAHO Potato :
10kg costs $5.78
Price per kg:
$5.78 / 10
= $0.578 per kg
0.518 < 0.578
Hence, PEI famous potato is a better deal
Rearrange t= 5s + 4 to make s the subject.
In a luck experiment the sample space is N = {1, 2, 3, 4]. We define the possibilities A = {1, 2}, B = {1, 3}, C = {1, 4}. If the elementary possibilities are equally probable, consider whether possibilities A, B, C are in pairs independently and if possibilities A, B, C are every three independently that is, completely independent.
Given,In a luck experiment the sample space is N = {1, 2, 3, 4]. We define the possibilities A = {1, 2}, B = {1, 3}, C = {1, 4}.
If the elementary possibilities are equally probable, we need to determine whether possibilities A, B, C are in pairs independently and if possibilities A, B, C are every three independently, i.e., completely independent.
An independent event is an event that is not affected by any other event or occurrence. When two events are independent, the probability of one event occurring does not affect the probability of the other event occurring.So, if we define three events, A, B, and C, then A and B, A and C, and B and C may be independent of each other, or they may be dependent on each other.
To determine whether they are independent or not, we need to find the probability of each event and its combinations.
Here, the probability of each elementary possibility is equally probable, i.e., 1/4.If we consider events A and B, then we see that they have 1 as their common element.
Hence, P(A and B) = P({1}) = 1/4.Now, P(A) = P({1, 2}) = 2/4 = 1/2, and P(B) = P({1, 3}) = 2/4 = 1/2.Then, P(A) × P(B) = (1/2) × (1/2) = 1/4 = P(A and B).Since P(A and B) = P(A) × P(B), we can say that events A and B are independent.Similarly, we can calculate for events A and C, and B and C. We get,P(A and C) = 1/4 = P(A) × P(C)P(B and C) = 1/4 = P(B) × P(C)Therefore, events A, B, and C are pairwise independent.
If events A, B, and C are completely independent, then their joint probability, i.e., P(A and B and C) is the product of their individual probabilities, i.e., P(A) × P(B) × P(C).If this holds, then A, B, and C are completely independent.
Now, we can calculate,P(A and B and C) = P({1}) = 1/4 = P(A) × P(B) × P(C)Since P(A and B and C) = P(A) × P(B) × P(C), we can say that events A, B, and C are completely independent.
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According to the given luck experiment, the events A, B, and C are all independent of each other.
The sample space is N = {1, 2, 3, 4}.
It is defined that the possibilities A = {1, 2}, B = {1, 3}, and C = {1, 4}.
If the elementary possibilities are equally probable, let's consider the independence of the possibilities A, B, and C as follows;
The event A and B are independent if and only if P(A ∩ B) = P(A)P(B).
Probability of A = P(A) = n(A) / n(S) = 2/4 = 1/2
Probability of B = P(B) = n(B) / n(S) = 2/4 = 1/2
Possibility of A ∩ B = {1}
P(A ∩ B) = n(A ∩ B) / n(S) = 1/4
Now, P(A)P(B) = (1/2) (1/2) = 1/4
Hence, P(A ∩ B) = P(A)P(B).
Therefore, the events A and B are independent.
The event A and C are independent if and only if P(A ∩ C) = P(A)P(C).
Probability of C = P(C) = n(C) / n(S) = 1/2
Probability of A = P(A) = n(A) / n(S) = 1/2
Possibility of A ∩ C = {1}
P(A ∩ C) = n(A ∩ C) / n(S) = 1/4
Now, P(A)P(C) = (1/2) (1/2) = 1/4
Therefore, P(A ∩ C) = P(A)P(C)
Thus, the events A and C are independent.
The event B and C are independent if and only if P(B ∩ C) = P(B)P(C).
Probability of B = P(B) = n(B) / n(S) = 1/2
Probability of C = P(C) = n(C) / n(S) = 1/2
Possibility of B ∩ C = {1}
P(B ∩ C) = n(B ∩ C) / n(S) = 1/4
Now, P(B)P(C) = (1/2) (1/2) = 1/4
Hence, P(B ∩ C) = P(B)P(C)
Thus, the events B and C are independent.
So, we have concluded that the events A, B, and C are all independent of each other.
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