Answer:
Exponential Decay or Growth
Step-by-step explanation:
[tex]y=C(1-r)^t[/tex] represents an exponential.
When r is greater than 0 but less than 1, the function decays by decreasing the values through the rate.
When r is less than 0 or negative, the function grows by increasing the values through the rate.
A box of Munchkins contains chocolate and glazed donut holes. If Jacob ate 2 chocolate
Munchkins, then 1/11 of the remaining Munchkins would be chocolate. If he instead added 4
glazed Munchkins to the original box, 1/7 of the Munchkins would be chocolate. How many total
Munchkins are in the original box?
Answer:
24 munchkins.
Step-by-step explanation:
Let C be the number of chocolate and D be number of glazed donut holes in the original box.
We are told if Jacob ate 2 chocolate munchkins, then 1/11 of the remaining Munchkins would be chocolate. We can represent this information as:
[tex]C-2=\frac{1}{11}*(C+D-2)...(1)[/tex]
We are also told if he instead added 4 glazed Munchkins to the original box, 1/7 of the Munchkins would be chocolate. We can represent this information as:
[tex]C=\frac{1}{7}*(C+D+4)...(2)[/tex]
Upon substituting C's value from equation (2) in equation (1) we will get,
[tex]\frac{1}{7}*(C+D+4)-2=\frac{1}{11}*(C+D-2)[/tex]
Let us have a common denominator on right side of equation.
[tex]\frac{1}{7}*(C+D+4)-\frac{7*2}{7}=\frac{1}{11}*(C+D-2)[/tex]
[tex]\frac{C+D+4-14}{7}=\frac{1}{11}*(C+D-2)[/tex]
Multiplying both sides of our equation by 7, we will get,
[tex]7*\frac{C+D-10}{7}=7*\frac{1}{11}*(C+D-2)[/tex]
[tex]C+D-10=\frac{7}{11}*(C+D-2)[/tex]
Multiplying both sides of our equation by 11, we will get,
[tex]11*(C+D-10)=11*\frac{7}{11}*(C+D-2)[/tex]
[tex]11*(C+D-10)=7*(C+D-2)[/tex]
[tex]11C+11D-110=7C+7D-14[/tex]
[tex]11C-7C+11D-7D=-14+110[/tex]
[tex]4C+4D=96[/tex]
[tex]4(C+D)=96[/tex]
[tex](C+D)=\frac{96}{4}[/tex]
[tex](C+D)=24[/tex]
Therefore, the total number of Munchkins in original box is 24.
How do i translate (x,y) ?
One of my questions is asking me to translate , (x,y) i know the formula for translating, (a,b) x+a,y+b But a=x and b=y. How do i solve this?
Answer: (x, y - 8)
Step-by-step explanation:
Start with the innermost parentheses (reflection over the y-axis), which changes the sign of the y-coordinate:
(x, y) → (-x, y)Now perform the next inner parenthesis (rotation 180°), which changes the signs of both the x- and y-coordinates:
(-x, y) → (x, -y)Now perform the last set of parenthesis (reflection over y=4), which has the rule of (x, y) → (x, y + 2(4 - y)) = (x, y + 8 - 2y) = (x, 8 - y):
(x, -y) → (x, -8 + y)Graph the image of this figure after a dilation with a scale factor of 1/3 centered at the origin.
Use the polygon tool to graph the dilated figure.
Answer:
see attached diagram
Step-by-step explanation:
A dilation with a scale factor of [tex]\dfrac{1}{3}[/tex] centered at the origin has a rule:
[tex](x,y)\mapsto\left(\dfrac{x}{3},\dfrac{y}{3}\right).[/tex]
Then the image of the triangle ABC is triangle DEF(see attached diagram) with coordinates
[tex]D\left(\dfrac{-3}{3},\dfrac{3}{3}\right)\Rightarrow D(-1,1);[/tex][tex]E\left(\dfrac{3}{3},\dfrac{9}{3}\right)\Rightarrow E(1,3);[/tex][tex]F\left(\dfrac{6}{3},\dfrac{6}{3}\right)\Rightarrow F(2,2).[/tex]What is the degree of this pynomial?
The given polynomial only has one exponent, so the degree of the polynomial is the number of the exponent.
The degree is 3.
Find the value of x in the trapezoid below. Show equations and all work that leads to your answer.
Answer:
x = 21.5
Step-by-step explanation:
In the given trapezoid, the angles are consecutive interior angles.
i.e, consider the parallel sides of a trapezoid as parallel lines and the common side as the transversal. So, it is parallel lines cut by a transversal make a pair of consecutive interior angles which are supplementary.
i.e,
(5x + 13)° + (3x - 5)° =180°
5x + 13 + 3x - 5 = 180
8x + 8 = 180
8x = 180 - 8 =172
x= 172/8 = 21.5
So, 5x + 13 = 5(21.5) + 13 = 120.5°
3x - 5 = 3(21.5) -5 = 59.5°
You buy 2 gallons of milk at $3.29/gallon, 4 gallons of ice cream at $3.59/gallon, and 5 lbs of strawberries at $0.72/lb. Food sales tax is 3.75%. What is your total cost?
Hi There!
Solution + Answer:
2 gallons of milk: 3.29 * 2 = $6.58
4 gallons of ice-cream: 3.59 * 4 = $14.36
5 pounds of strawberries: 0.72 * 5 = $3.60
Total price before tax: 6.58 + 14.36 + 3.60 = $24.54
Tax: 24.54 * 0.0375 = $0.92025
Total price after tax: 24.54 + 0.92025 = $25.46025
Round: 25.46025 = $25.46
Hope This Helps :)
According to the question,
2 gallons of milk,
= [tex]3.29\times 2[/tex]
= [tex]6.58[/tex] ($)
4 gallons of ice-cream,
= [tex]3.59\times 4[/tex]
= [tex]14.36[/tex] ($)
5 pounds of strawberries,
= [tex]0.72\times 5[/tex]
= [tex]3.60[/tex] ($)
Now,
The total price before tax will be:
= [tex]6.58+14.36+3.60[/tex]
= [tex]24.54[/tex] ($)
The tax will be:
= [tex]24.54\times 0.0375[/tex]
= [tex]0.92025[/tex] ($)
hence
The total price after tax will be:
= [tex]24.54+0.92025[/tex]
= [tex]25.46025[/tex]
or,
= [tex]25.46[/tex] ($)
Learn more about sales tax here:
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Can you please put the answer to 60000000x1500 because I have a really important test cin=ming up and
What is the value of the discriminant of the quadratic equation -2x^2=-8x+8, and what does its value mean about the number of real number solutions the equation has?
Answer:
B on edg
Step-by-step explanation:
You got this chief.
Which function represents a line with a slope of ?4 and a y-intercept of ?2? A) y = 4x ? 2 B) y = ?4x + 2 Eliminate C) y = ?4x ? 2 D) y = ?2x ? 4
Answer:
A function describes this way is y = 4x + 2
Step-by-step explanation:
In order to find this, we have to start with slope-intercept form.
y = mx + b
Knowing this, we can then input the slope in for m and the intercept in for b. This will give us the equation.
y = 4x + 2
There are more inches on a football field (100 yards) than there are centimeters on a 28 meter basketball court. True or false?
Answer:
true
Step-by-step explanation:
on a football feild there are 4,320 inches and on a basketball court there are 2,865 centemeters
Lisa spent 8 minutes on the phone while routing 2 phone calls. In all, how many phone calls does Lisa have to route to spend a total of 12 minutes on the phone? Solve using unit rates.
Lisa spends 4 minutes per call, so to spend a total of 12 minutes on the phone, she needs to route 3 calls.
Lisa spent 8 minutes on the phone routing 2 phone calls, which means she spends 4 minutes per call. This is calculated using the unit rate which is the time spent per phone call. To find out how many phone calls Lisa would have to route to spend a total of 12 minutes, we can set up a proportion where 4 minutes corresponds to 1 phone call (the unit rate), and 12 minutes corresponds to the number of phone calls we want to find (let's call it x calls).
The proportion is as follows:
4 minutes/1 call = 12 minutes/x calls
Cross-multiplying to solve for x gives us:
4 * x = 12 * 1
4x = 12
x = 12 / 4
x = 3 calls
Therefore, Lisa needs to route a total of 3 calls to spend 12 minutes on the phone.
Please answer this question for me!! Forty points and brainliest!
Answer: The Answer is B
Step-by-step explanation:
FIrst we see post 7 and -5 we need to cancel out seven sooooo we make sevena a neg and do -5 and -7 and we get -12 we divide this by -2 and we get post 6
x=6
-2x+7=-5
-2(-7)+7=-5(-7)
-2(-2)x=-12(-2)
x=6
Answer:
b 6
Step-by-step explanation:
-2x+7 = -5
Subtract 7 from each side
-2x +7-7 = -5 -7
-2x = -12
Divide by -2
-2x/2 = -12/-2
x=6
The ordered pairs model an exponential growth function. {(−1,34), (0,51), (1,76.5), (2,114.75)} What is the function equation? f(x)=__________
Answer:
[tex]y= 51(1.5)^x[/tex]
Step-by-step explanation:
The ordered pairs model an exponential growth function. {(−1,34), (0,51), (1,76.5), (2,114.75)}
Exponential function equation is y=ab^x
Lets plug in the ordered pairs and find our 'a' and 'b'
(0,51)
[tex]51= ab^0[/tex]
a= 51
(−1,34)
[tex]34= 51b^{-1}[/tex]
Divide both sides by 51
[tex]\frac{34}{51} = b^{-1}[/tex]
So b = 51/34
b= 1.5
Plug in the values of 'a' and 'b'
Equation becomes [tex]y= 51(1.5)^x[/tex]
contractors insurance has a $75 sign-up fee. Once singed up the insurace cost $61 each month. Which of the following equations represents the relationship between the number of months a contractor is insured,m, and the total cost of contractors insurance,c?
Answer:
The equation should read c = 61m + 75
Step-by-step explanation:
We can find this because we know that the 61 dollars is a dependent cost on the number of months. Therefore they have to be multiplied together.
The 75 is a constant, which means it gets added to the end.
15 rulers cost £3
How much do 40 rulers cost?
Answer:
The cost of the 40 rulers is £8 .
Step-by-step explanation:
As given in the question
15 rulers cost £3
i.e
15 rulers = £3
Cost of the one ruler.
[tex]1\ ruler = \frac{3}{15}[/tex]
1 ruler cost = £ 0.2
Now find out the cost for 40 rulers .
Cost for 40 rulers = 40 × 0.2
= £8
Therefore the cost of the 40 rulers is £8 .
The question involves finding the unit price of a ruler and then finding the cost of 40 rulers. The price for one ruler is £0.2, and hence, the price for 40 rulers would be £8.
Explanation:This is a mathematics question about unit price and proportionality. First, we need to find the price of one ruler by dividing the total cost by the number of rulers. So, £3 divided by 15 rulers is £0.2 per ruler. To find out how much 40 rulers costs, we multiply the price of one ruler by the quantity we desire which is 40 rulers. So, £0.2 multiplied by 40 equals to £8.
Therefore, the cost of 40 rulers would be £8.
Learn more about Unit Price and Proportionality here:
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#SPJ3
Help me with this question! Anybody. {will receive 15 pts.}
Answer:
The answer is 5.
Step-by-step explanation:
5+(-9) or
5-9=-4
What is the slope of the line? – –Which equation shows the point-slope form of the line that passes through (3, 2) and has a slope of ? y + 2 =(x + 3) y – 2 = (x – 3) y + 3 = (x + 2) y – 3 = (x – 2)
Answer:
The correct answer is B) y – 2 = (x – 3)
Step-by-step explanation:
In order to find this, look at the base form of point-slope form.
y - y1 = m(x - x1)
Seeing as there is subtraction symbols after x and y, we know that we would be subtracting the x and y values. Since 2 is the y value, it would go in for y1 and 3 being the x value puts it in for x1. Once you do that, you'll get the equation above.
Answer:
-1/2
Step-by-step explanation:
The probability of an event is 3/10 . What are the odds of the same event?
Choices:
10/13
3/13
7/10
3/7
Please explain your answer if possible. Thanks
Answer:
D. [tex]\frac{3}{7}[/tex]
Step-by-step explanation:
We have been given that the probability of an event is 3/10.
To find the odds of the same event we will use formula:
[tex]\text{Odds of an event}=\frac{\text{Probability of the event}}{\text{1-Probability of the event}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{1-\frac{3}{10}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{\frac{1*10}{10}-\frac{3}{10}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{\frac{10-3}{10}}[/tex]
[tex]\text{Odds of the event}=\frac{\frac{3}{10}}{\frac{7}{10}}[/tex]
Dividing a fraction with another fraction is same as multiplying the 1st fraction by the reciprocal of second fraction.
[tex]\text{Odds of the event}=\frac{3}{10}\times \frac{10}{7}[/tex]
[tex]\text{Odds of the event}=\frac{3}{7}[/tex]
Therefore, the odds of the same event is [tex]\frac{3}{7}[/tex] and option D is the correct choice.
A construction company plans to invest in a building project. There is a 30% chance that the company will lose $30,000, a 40% chance of a break even, and a 30% chance of a $60,000 profit. Based ONLY on this information, what should the company do?
The expected value is $9,000.00, so the company should proceed with the project.
B) The expected value is $18,000.00, so the company should proceed with the project.
C) The expected value is −$9,000.00, so the company should not proceed with the project.
D) The expected value is −$18,000.00, so the company should not proceed with the project.
Answer: Option 'A' is correct.
Step-by-step explanation:
Since we have given that
30% chance that the company will lose $30000.
40% chance of a break even that there is no loss and no profit.
30% chance that the company will profit $ 60000.
As we know the formula for "Expectation":
So, Expected value will be
[tex]\frac{30}{100}\times (-30000)+\frac{40}{100}\times 0+\frac{30}{100}\times 60000\\\\=03\times (-30000)+0.4\times 0+0.3\times 60000\\\\=-9000+18000\\\\=\$9000[/tex]
Expected value is $9000. So, the company should proceed with the project.
Hence, Option 'A' is correct.
What is a disadvantage of using credit?
A. It can be tempting to overspend.
B. It’s convenient.
C. It may offer special advantages.
D. All of the above
Final answer:
A disadvantage of using credit is that it can be tempting to overspend.
Explanation:
A disadvantage of using credit is that it can be tempting to overspend.
When using credit, individuals can easily spend more than they can afford, leading to financial difficulties and debt. It is important to use credit responsibly and only borrow what can be comfortably repaid.
Using knowledge that you have gained throughout this lab, what is the horizontal asymptote of y=(2x+3)(5x-2)/7x^2-3
Answer:
(no horizontal asymptotes found)
Step-by-step explanation:
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Answer:
Step-by-step explanation:
Problem One (left panel)
Question A
The y intercept happens when x = 0That being said, the y intercept is 50. It was moving when the timing began.Question B
The rate of change = (56 - 52)/(3 - 1) = 4/2 = 2 miles / hour^2 (you have a slight acceleration.
Question C
60 = a + (n-1)d60 = 50 + (n - 1)*210/2 = (n - 1)*2/25 = n - 1 6 = nThe way I have done it the domain is n from 1 to 6
Question 2 (Right Panel)
Question A
The equation for the table is f(x) = 3x - 3 which was derived simply by putting all three points into y = ax + b and solving.
f(0) = ax + b-3 = a*0) + bb = - 3So far what you have is f(x) = ax - 3f(-1) = a*(-1) - 3 but we know (f(-1)) = -6- 6 = a(-1) - 3 add 3 to both sides-6 +3 = a(-1) -3 + 3-3 = a*(-1) Divide by - 1a = 3f(x) = 3x - 3 Answer for f(x)The slope of f(x) = the coefficient in front of the xf(x) has a slope of 3g(x) has a slope of 4Part B
f(x) has a y intercept of - 3g(x) has a y intercept of -5f(x) has the greater y intercept.-3 > - 5 I answered 5 questions that are EASY AS FUDGE LOL and i just want someones feedback to see if it is right WILL GIVE BRAINLIEST HELPPPPPPPPPPPPP at least take a look at it!
1. The number of people that belong to a certain social website is 412 and growing at a rate of 5% every month.
How many members will there be in 9 months?
Enter your answer, rounded to the nearest whole number, in the box.
MY ANSWER IS: 639......RIGHT OR WRONG
2. The expression 15.85(1+x) gives the total cost of a toy, where x is the sales tax written in decimal form.
What does 1 + x represent in the expression?
percent of original price being paid in tax
cost of toy
percent of tax
amount of tax
3. A population of lizards decreases exponentially at a rate of 12.5% per year.
What is the equivalent monthly rate to the nearest hundredth of a percent?
0.20%
1.04%
1.11%
1.23%
4. Enrollment was increasing at a rate of 20% per year.
What was the monthly growth rate?
Enter your answer, rounded to the nearest tenth of a percent, in the box.
MY ANSWER IS: 1.7%
5. The value of a truck after t years is represented by the function f(t)=22,200(0.92)t .
What does the value 22,200 represent in this situation?
The value of the truck increases by $22,200 each year.
The initial value of the truck is $22,200.
The value of the truck decreases by $22,200 each year.
The value of the truck after t years is $22,200.
Answer:
i got 185.4 so basically 184.
412 times 5% equals 20.6
times that by 9 for 9 months then that gives you the answer.
Answer:
1. The population increases exponentially, therefore the number of members in the team will increase by the equation modeled by :
[tex]\text{Number of members = }a_0(1+\frac{Rate}{100})^{Time}\\\\\implies\text{Number of members = }412(1+\frac{5}{100})^9\\\\\implioes\text{Number of members after 9 months = }639[/tex]
2. Total cost of the toy is calculated by multiplying the initial cost and the amount of tax charged on the toy.
Here, 15.85 represents the initial cost of the toy
Hence, 1 + x represents the amount of tax charged on the toy.
3. Rate per year is given to be 12.5%
Now, to find monthly rate, divide the yearly rate by 12
[tex]\text{Monthly Rate = }\frac{12.5}{12}=1.04\%[/tex]
4. Rate per year is given to be 20%
Now, to find monthly rate, divide the yearly rate by 12
[tex]\text{Monthly Rate = }\frac{20}{12}\approx 1.7\%[/tex]
5. The value of a truck after t years is represented by the function f(t)=22,200(0.92)t .
Here, 2200 is the initial value of the truck.
What are the coordinates of the midsegment that is parallel to side BC.
(0, 2) and (1, 0)
(0, 2) and (-2, -1)
(1, 0) and (2, 1)
(1, 0) and (-2, -1)
Answer:
Correct choice is D (1,0) and (-2,-1)
Step-by-step explanation:
Triangle DBC has vertices B(-3,1), C(3,3) and D(-1,-3). By the triangle midline theorem, the midline joining the midpoints of two sides is parallel to the third side. Thus, you have to find the coordinates of the midpoints of the sides DB and DC of the triangle DBC.
Let point E be the midpoint of the side BD, then point E has coordinates [tex]\left(\dfrac{x_B+x_D}{2},\dfrac{y_B+y_D}{2}\right).[/tex] If B(-3,1) and D(-1,-3), then
[tex]E\left(\dfrac{-3+(-1)}{2},\dfrac{1+(-3)}{2}\right)\Rightarrow E(-2,-1).[/tex]
Let point F be the midpoint of the side DC, then point F has coordinates [tex]\left(\dfrac{x_C+x_D}{2},\dfrac{y_C+y_D}{2}\right).[/tex] If C(3,3) and D(-1,-3), then
[tex]F\left(\dfrac{3+(-1)}{2},\dfrac{3+(-3)}{2}\right)\Rightarrow F(1,0).[/tex]
Answer:
D: (1,0) and (-2,-1)
Step-by-step explanation:
if you where to plot all the other answers and take a straight edge (ruler) and drew a line through them you would find this is the only proper answer. and i used geogebra
Suppose the heights of the members of a population follow a normal distribution. If the mean height of the population is 68 inches and the standard deviation is 4 inches, 95% of the population will have a height within which of the following ranges?
The empirical rule states that at 95% the measurements would be within 2 standard deviations of the mean.
You are given a mean of 68 inches and a standard deviation of 4.
2 times the standard deviation = 2 x 4 = 8
So 95% of the heights would be between 68-8 = 60 inches and 68+8 = 76 inches.
Answer:
60 -76 is the range
Step-by-step explanation:
As the graph shows, if we are in 2 standard deviations of the mean, we are in (34.1 + 13.6) *2 = 47.7*2 = 95.4 %
Our mean is 68
2 standard deviations is 2 * 4 = 8
68-8 = 60
68 * 8 = 76
We need to be between 60 and 76 to have at 95% confidence interval
Find the missing side length. Round to the nearest tenth if needed. 11.7 is incorrect! PLEASE HELP!!
Hello from MrBillDoesMath!
Answer:
10.2
Discussion:
Let the length of the missing side = "s" Applying the Pythagorean theorem to this right triangle gives:
4^2 + s^2 = 11^2 =>
16 + s^2 = 121 => ( 4*4 = 16, 11*11 = 121)
16 - 16 + s^2 = 121 - 16 => (subtract 16 from both sides)
s^2 = 105 =>
x = sqrt (105)
which is approximately 10.2
Regards,
MrB
P.S. I'll be on vacation from Friday, Dec 22 to Jan 2, 2019. Have a Great New Year!
All of the questions below are regarding the attached photo. Please do not answer if you're not gonna answer all of them. (VALID ANSWERS OF COURSE)
a. What is the slope of the line?
b. What does the slope represent?
c. What type of slope is represented by the graph?
d. What is the -intercept?
e. Define the variables x and y.
f. Write a function, f (x), that represents this situation.
Answer:
The slope is 2
The slope represents how much it costs per visit the pool.
We have a positive slope when lines go from bottom left to top right
The y intercept is where we cross the y axis which is 4.
x is the number of visits to the pool
y is the total cost in dollars
f(x) = 2x+4
Step-by-step explanation:
To find the slope we take two point ( 0,4) and (1,6)
and use the formula
m = (y2-y1)/(x2-x1)
= (6-4)/(1-0)
m= 2/1
The slope is 2
It represents how much it costs per visit the pool.
We have a positive slope when lines go from bottom left to top right
The y intercept is where we cross the y axis which is 4.
x is the number of visits to the pool
y is the total cost in dollars
f(x) = mx+b where m is the slope and b is the y intercept
f(x) = 2x+4
Answer:
the answer is D edge 2021
Step-by-step explanation:
Ben is 20 years older than Daniel. Ben and Daniel first met two years ago. Three years ago, Ben was 33 times as old as Daniel.
how old is Ben now?
Answer: 33
Step-by-step explanation:
Today 3 years ago
Ben: x + 20 (x + 20) - 3
Dan: x x - 3
3 years ago, Ben was 3 times as old as Daniel
⇒ (x + 20) - 3 = 3(x - 3)
x + 17 = 3x - 9
17 = 2x - 9
26 = 2x
13 = x
Today, Ben = x + 20
= 13 + 20
= 33
Answer:
Ben is 33 years old
Step-by-step explanation:
Let Ben’s age = B
Let Daniel’s age = x
B = x + 20 (20 years older than Daniel)
Their ages 3 years ago:
B = 3x (also given in problem)
B = x + 20 (given in problem)
x + 20 = 3x (Since both equations are equal to B, set them equal to each other)
20 = 2x (simplify)
x = 10
B = 10 + 20 = 30
Their ages now:
Add 3 to their "ages three years ago"
New x = Daniel’s age = old age + 3 = 10 + 3 = 13 years old
B = Ben’s age = old age + 3 = 30 + 3 = 33 years old
PLEASE HELP ASAP!!! CORRECT ANSWERS ONLY PLEASE!!
Solve: log2(x-4) = 4
Answer: D
Step-by-step explanation:
log₂ (x - 4) = 4
x - 4 = 2⁴
x - 4 = 16
x = 20
Two events E1 and E2 are called independent if p(E1 â© E2) = p(E1)p(E2). For each of the following pairs of events, which are subsets of the set of all possible outcomes when a coin is tossed three times, determine whether or not they are independent. a) E1: tails comes up with the coin is tossed the first time; E2: heads comes up when the coin is tossed the second time. b) E1: the first coin comes up tails; E2: two, and not three, heads come up in a row. c) E1: the second coin comes up tails; E2: two, and not three, heads come up in a row.
Answer: a) Independent
b) Independent
c) Dependent
Step-by-step explanation:
Since, If a coin is tossed three times,
Then, total number of outcomes, n(S) = 8
a) [tex]E_1[/tex] : tails comes up with the coin is tossed the first time;
[tex]E_1[/tex] = { TTT, THH, THT, TTH }
[tex]E_2[/tex] : heads comes up when the coin is tossed the second time.
[tex]E_2[/tex] = { THT, HHH, THH, HHT }
Thus, [tex]n(E_1)=4[/tex]
⇒ [tex]P(E_1)=\frac{n(E_1)}{n(S)}=\frac{4}{8}=\frac{1}{2} [/tex]
Similarly, [tex]P(E_2)=\frac{1}{2}[/tex]
⇒ [tex]P(E_1)\times P(E_2)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4} [/tex]
Since, [tex]E_1\cap E_2 [/tex] = { THH, THT }
[tex]n(E_1\cap E_2) = 2 [/tex]
⇒ [tex]P(E_1\cap E_2) = \frac{n(E_1\cap E_2)}{n(S)}= \frac{2}{8}=\frac{1}{4}[/tex]
Thus, [tex]P(E_1\cap E_2)=P(E_1)\timesP(E_2)[/tex]
Therefore, [tex]E_1[/tex] and [tex]E_2[/tex] are independent events.
B) [tex]E_1[/tex] : the first coin comes up tails
[tex]E_1[/tex] = { TTT, THH, THT, TTH }
[tex]E_2[/tex] : two, and not three, heads come up in a row
[tex]E_2[/tex] = { HHT, THH }
Thus, [tex]n(E_1)=4[/tex]
⇒ [tex]P(E_1)=\frac{n(E_1)}{n(S)}=\frac{4}{8}=\frac{1}{2} [/tex]
Similarly, [tex]P(E_2)=\frac{1}{4}[/tex]
⇒ [tex]P(E_1)\times P(E_2)=\frac{1}{2}\times \frac{1}{4}=\frac{1}{8} [/tex]
Since, [tex]E_1\cap E_2 [/tex] = { THH }
[tex]n(E_1\cap E_2) = 1 [/tex]
⇒ [tex]P(E_1\cap E_2) = \frac{n(E_1\cap E_2)}{n(S)}= \frac{1}{8}[/tex]
Thus, [tex]P(E_1\cap E_2)=P(E_1)\timesP(E_2)[/tex]
Therefore, [tex]E_1[/tex] and [tex]E_2[/tex] are independent events.
C) [tex]E_1[/tex] : the second coin comes up tails;
[tex]E_1[/tex] = { HTH, HTT, TTT, TTH }
[tex]E_2[/tex] : two, and not three, heads come up in a row
[tex]E_2[/tex] = { HHT, THH }
Thus, [tex]n(E_1)=4[/tex]
⇒ [tex]P(E_1)=\frac{n(E_1)}{n(S)}=\frac{4}{8}=\frac{1}{2} [/tex]
Similarly, [tex]P(E_2)=\frac{1}{4}[/tex]
⇒ [tex]P(E_1)\times P(E_2)=\frac{1}{2}\times \frac{1}{4}=\frac{1}{8} [/tex]
Since, [tex]E_1\cap E_2 [/tex] = [tex]\phi[/tex]
[tex]n(E_1\cap E_2) = 0 [/tex]
⇒ [tex]P(E_1\cap E_2) = 0[/tex]
Thus, [tex]P(E_1\cap E_2)\neq P(E_1)\timesP(E_2)[/tex]
Therefore, [tex]E_1[/tex] and [tex]E_2[/tex] are dependent events.
Answer:
P (E1) = 18 / 38
P (E2) = 18 / 38
P (E1 and E2) = 10 / 38
Step-by-step explanation: