The most probable number of heads becomes more and more likely as the number of tosses increases.
Let's denote the probability of observing tails as q (which is 1/2 for a fair coin). Then the probability of observing exactly n heads in 4n tosses is given by the binomial distribution:
p(n) = (4n choose n) * (1/2)^(4n)
where (4n choose n) is the number of ways to choose n heads out of 4n tosses. We can express this in terms of the most probable number of heads, which is 2n:
p(n) = (4n choose n) * (1/2)^(4n) * (2^(2n))/(2^(2n))
= (4n choose 2n) * (1/4)^n * 2^(2n)
where we used the identity (4n choose n) = (4n choose 2n) * (1/4)^n * 2^(2n). This identity follows from the fact that we can choose 2n heads out of 4n tosses by first choosing n heads out of the first 2n tosses, and then choosing the remaining n heads out of the last 2n tosses.
Now we can express the ratio p(n)/p(2n) as:
p(n)/p(2n) = [(4n choose 2n) * (1/4)^n * 2^(2n)] / [(4n choose 4n) * (1/4)^(2n) * 2^(4n)]
= [(4n)! / (2n)!^2 / 2^(2n)] / [(4n)! / (4n)! / 2^(4n)]
= [(2n)! / (n!)^2] / 2^(2n)
= (2n-1)!! / (n!)^2 / 2^n
where (2n-1)!! is the double factorial of 2n-1. Note that (2n-1)!! is the product of all odd integers from 1 to 2n-1, which is always less than or equal to the product of all integers from 1 to n, which is n!. Therefore,
p(n)/p(2n) = (2n-1)!! / (n!)^2 / 2^n <= n! / (n!)^2 / 2^n = 1/(n * 2^n)
As n increases, the denominator n * 2^n grows much faster than the numerator (2n-1)!!, so the ratio p(n)/p(2n) approaches zero. This means that the probability of observing n heads relative to the most probable number of heads becomes vanishingly small as n increases, which is consistent with the intuition that the most probable number of heads becomes more and more likely as the number of tosses increases.
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I need help on number 7
Answer:
y=2x+2 and increasing
Step-by-step explanation:
we see there is a pint (0,2) so the y intercept is 2, we know the slope is 2 because the slope formula can be used on ay two points, for example (0,2) and (1,4) 4-2 is 2 and 1-0 is 1. 2/1 =2 so the slope is 2
Rewrite the function by completing the square.
f(x) = x^2 + 14x + 64
Step-by-step explanation:
step 1. (x^2 + 14x + 49) + 15
step 2. (x + 7)^2 + 15.
5. A jewelry supply shop buys silver chains from a manufacturer for $45.00. and then sells the
chains at a 57% markup. What is the retail price of a silver chain?
Which expressions are equivalent to 8(-10x+3.5y-7)? Select two options.
Answer:
[tex] - 8x + 120y - 56[/tex]
Step-by-step explanation:
evaluate the power calculate
[tex]8(1 x + 3 \times 5y - 7[/tex]
the coefficient of 1 doest have to be written but keep the sign
[tex]8( - 1x + 15y - 7[/tex]
remove the parentheses
Other option
D: 15.7
Answer:
I see a typo in the problem. When it says LR. This triangle isn't in the direction of L its the opposite direction. I think the answer might be C but im not completely sure.
Step-by-step explanation:
Big Brain
what is one astronomical unit in inches
Answer:
5.89E+12
Step-by-step explanation:
Assume that the lines are parallel.
Answer:
left ?= 110
right?= 70
Step-by-step explanation:
Write the answer in its simplest form. -8 + 13
Answer:
5
Step-by-step explanation:
-8 + 13 is equal to 5. It is positive since the larger number is positive.
A recipe requires 4 cups of flour to make 28 dinner rolls. How many cups of flour is needed to make 112 rolls?
Answer:
16 cups
Step-by-step explanation:
You have the ratio 4 cups = 28 dinner rolls. 28 * 4 = 112, so you know that if you multiply both sides of that equation by 4 you will get the number of cups of flout to make 28 dinner rolls:
4 cups = 28 dinner rolls
becomes
16 cups = 112 dinner rolls
so the answer is 16 cups
Answer:
16 cups of flour are needed to make 112 dinner rolls.
Step-by-step explanation:
28 divided by 4 is 7 which is the number of rolls for one cups of flour. Divide 112 by 7 and thats the amount of flour needed for 112 dinner rolls. You can also multiply 28 4 times since 4 can go into 16 when multiplied.
Rashid can do a piece of work in 8 days, which Tipu can finish in 12 days. If they work on alternate days with Rashid Beginning, in how many days, the work will be finished?
Answer: 9.6 days
Step-by-step explanation:
Rashid can do a piece of work in 8 days. Rashid work rate will then be:
= 1/8 owe day.
Tipu can finish in 12 days. Tipu's work rate will be: = 1/12 owe day.
Their work rate together will then be:
= 1/8 + 1/12
= 3/24 + 2/24
= 5/24
From the calculation, together they will finish the work in:
= 1 ÷ 5/24
= 1 × 24/5
= 24/5 days
= 4.8 days
Busy since they are working on alternate days, then this will be multiplied by 2 which will be:
= 4.8 × 2
= 9.6 days
A TV has a listed price of $578.98 before tax. If the sales tax rate is 8.25%, find the total cost of the TV with sales tax included.
Round your answer to the nearest cent, as necessary.
Answer:
$626.75
Step-by-step explanation:
First, we must find how much tax will be added to the TV's price:
Step 1: Convert the percentage to a decimal:
To convert a percentage to a decimal, divide it by 100
8.25/100 = 0.0825
Step 2: Multiply the TV's price by that decimal:
578.98 * 0.0825 = 47.76585
So $47.76585 will be added to the price
Now, we can calculate the total price:
Step 3: Add the tax we jsut calculated to the TV's price to find total price:
578.98 + 47.76585
= 626.74585
Step 4: Round to the nearest cent (2 decimal places)
626.75
So the total price is $626.75
If f(x) = x2, and
g(x) = x - 1, then
f(g(x)) = ?
Answer:
f(g(x))=1x^2 +-2x+1
Step-by-step explanation:
your inputs should be
1
-2
1
If f(x) = [tex]x^{2}[/tex], g(x) = [tex]x -1[/tex] then f(g(x)) = [tex]1x^{2} -2x+1[/tex].
What is a function?Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given
f(x) = [tex]x^{2}[/tex]
g(x) = [tex]x -1[/tex]
f(g(x)) = [tex](x-1)^{2}[/tex]
It is in the form of [tex](a-b)^{2}=a^{2} -2(a)(b) +b^{2}[/tex]
= [tex]x^{2} -2(x)(1) +1^{2}[/tex]
= [tex]x^{2} -2x+1[/tex]
f(g(x)) = [tex]1x^{2} -2x+1[/tex]
Hence, If f(x) = [tex]x^{2}[/tex], g(x) = [tex]x -1[/tex] then f(g(x)) = [tex]1x^{2} -2x+1[/tex].
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2 Do the following calculations
a) 55,8965 + 12,443 x 3
b) (6 x 3) - 4 + 26 - 2
c) 44,65 = 3 x 11,11 + 68
d) 24 - 16 + 3 + 5 - 6 + 10 - 2
e) 15 = (2+2)
Answer:
a) 93,2255
b)38
c)false
d)18
e)false
At midnight in Minnesota, the temperature was -59. Over the next three hours, the
temperature dropped 7. What was the temperature at 3:00 a.m.?
A-12
B-5°
C -2°
D 2
PLEASE HELP!!!
9514 1404 393
Answer:
A. -12°
Step-by-step explanation:
The final value is found by adding the change to the initial value.
final temp = initial temp + change in temp
final temp = -5° + (-7°) = (-5 -7)°
3 am temp = -12°
Answer:
A
Step-by-step explanation:
1. Write the equation: -5-7. We get that equation because we start at -5 and then it drops 7 degrees. "dropped" is the keyword that tells us to subtract.
2. Solve the equation: -12 degrees. You can solve it very easily by plugging it into a calculator or mentally.
P.S: You typed the question wrong, you said 56 degrees instead of 5. That really had me stumbled lol.
Answer this to be Donkey Kong's friend
36x + 324 - 28x + 196
combine like terms: 36x - 28x = 8x and 324 + 196 = 520
8x+520
The area covered by a topic lie plant triples every year. By what factors does the are covered by the plant increase every month?
Answer:
16.67%
Step-by-step explanation:
If area triples every year :
This means that :
With an initial area of 10;
Area at the end of the year = 3 * 10 = 30
Percentage increase per year :
(30 - 10) / 10 * 100%
(20 / 10) * 100%
2 * 100%
= 200% per year
Therefore, monthly increase %
Yearly % increase ÷ number of months in a year
200% ÷ 12
= 16.666666%
= 16.67%
How many gallons of gas would it take to drive 13,500 miles? (URGENT)
Answer:
24 gallons :)
Step-by-step explanation:
The graph shows your elevation during a run. Which statement is true about your run?
A- You were running uphill for the first 15 minutes.
B- You were running uphill between 10 and 20 minutes into the run.
C- You were running downhill for the last 10 minutea.
D- You were running downhill the first 15 minutes.
Answer:
Its D. You were running downhill the first 15 minutes.
What is the measure of arc BD?
Step-by-step explanation:
step 1. mBD = 63°. The arc angle measures the same from the center of a circle.
3/8÷5/2
..
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Answer:
3/20
Step-by-step explanation:
y=4x+6 2y=8x+12 what’s the solution is it one solution, no solution, or infinitely many solutions
Answer:
Infinitely many solutions.
Step-by-step explanation:
y=4x+6
2y=8x+12
------------
simplify 2y=8x+12 into y=4x+6
y=4x+6 matches the second equation.
x² + y² = 49
Note: Each column in the table represents an ordered pair. If multiple solutions exist, you only need to identify one.
Answer:
1....y=7
2...x=7
3...y=5
4...y=6
5...x=2√10
Can you help me please
Answer:
D?
Step-by-step explanation:
if i got it wrong im truly sorry
Is the point (3, 11) a solution for the linear equation 1/3x + y = 12? Explain.
Answer:
Yes, because when the values are substituted in, the equation becomes a true statement
Step-by-step explanation:
plug in the values for x and y and see if the equation makes sense
x=3
y=11
1/3(3)+11=12
1+11=12
12=12
Since the above equation is true, the point is a solution for the linear equation.
Find the measure of arc BC
Answer:
The measure of the Arc BC is 80
mean median mode and range,4,8,6,4,4,1,1
A shadow is cast by a pole. The angle of elevation from the end of the shadow to the top of
the pole is 35 degrees. If the distance from the base of the pole to the end of the shadow is 22
feet, how far is it from the end of the shadow to the top of the pole?
Answer:
13ft
Step-by-step explanation:
35-22=13
Ronald bought 1.5 pounds of red apples and 0.75 pounds of yellow apples. He bought 1.6 times as many pounds of green apples as red apples and yellow apples combined.
How many pounds of green apples did Ronald buy?
Answer:
To find the number of green apples he bought, We do the following:
calculate the total mass of red and yellow apples bought 1.5+0.75=2.25 poundsSince the mass of green apples bought was 1.6 times the combined mass of yellow and red apples bought we get;2.25×1.6=3.6pounds He bought 3.6 pounds green appleswhat is the equation of a line that passes through points (0,7) and( 2,8)write your answer in slope intercept form
Answer:
(0,7) and (2,8)
slope =8-7=1
2-0=2
m=1/2 meaning slope
equation=y-7/x-o=1/2
2(y-7)=1(x-0)
2y-14=x -o
2y=1x +14
y =1/2 x +7
Step-by-step explanation:
to get the slope is y 2-y 1 divide by x 2-x 1 to get 1/2
you then take one point as (x,y) and one of the coordinates =the slope
then you divide by the 2 on both sides in order to get y=mx + c
Factor the expression below.
x2 - 81
O A. (X-9)(x - 9)
O B. (x + 27)(x-3)
O C. (x-27)(x - 3)
O D. (x + 9)(x-9)