Answer:
The easiest way to figure this is to find P(no girls in 5 births) = (1/2)^5 = 1/32 -------------- The P(at least one girl in 5 births) = 1 - 1/32 = 31/32
The probability that when they have five children, at least one of them is a boy is 31/32
Probability is the likelihood or chance that an event will occur.
Probability = Expected outcome/Total outcome:
The probability that a couple will have a boy or girl among the five children = 1/2
If they gave birth to 5 children:
The probability that when they have five children, at least one of them is a boy = 1 - (1/2)^n where:
n is the total number of children
The probability that when they have five children, at least one of them is a boy = 1 - (1/2)ⁿ
The probability that when they have five children, at least one of them is a boy = 1 - (1/2)⁵
The probability that when they have five children, at least one of them is a boy = 1- 1/32 = 31/32
Hence the probability that when they have five children, at least one of them is a boy is 31/32
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Recall that the Fibonacci Sequence is defined by the recurrence relation, a0 = a1 = 1 and for n ≥ 2, an = an−1 + an−2 . a. Show that f(x) = 1 1−x−x 2 is the generating function of the Fibonacci Sequence. b. Find ???? and β such that 1 − x − x 2 = (1 − ????x)(1 − βx). c. Find A and B in terms of ???? and β, such that 1 1−x−x 2 = A 1−????x + B 1−βx. d. Use the results of the previous parts to obtain a formula for an.
Answer:
Step-by-step explanation:
From the given information:
[tex]a_n = a_{n-1} + a_{n-2}; \ \ \ n \ge 2 \\ \\ a_o = 1 \\ \\ a_1 =1 \ \ \ \ \ since \ \ a_o = a_1 = 1[/tex]
A)
[tex]a_n - a_{n-1} - a_{n-2} = 0 \\ \\ \implies \sum \limits ^{\infty}_{n=2}(a_n -a_{n-1}-a_{n-2} ) x^n = 0 \\ \\ \implies \sum \limits ^{\infty}_{n=2} a_nx^n - \sum \limits ^{\infty}_{n=2} a_{n-1}x^n - \sum \limits ^{\infty}_{n=2}a_{n-2} x^n = 0 \\ \\ \implies (a(x) -a_o-a_1x) - (x(a(x) -a_o)) -x^2a(x) = 0 \\ \\ \implies a(x) (1 -x-x^2) -a_o-a_1x+a_ox = 0 \\ \\ \implies a(x)(1-x-x^2)-1-x+x=0 \\ \\ \implies a(x) (1-x-x^2) = 1[/tex]
[tex]\mathbf{Generating \ Function: a(x) = \dfrac{1}{1-x-x^2}=f(x)}[/tex]
B)
[tex]If \ \ 1 -x-x^2 = (1 - \alpha x) ( 1- \beta x) \\ \\ \implies 1 -x - ^2 = 1 + \alpha \beta x^2 - ( \alpha + \beta )x \\ \\ \text{It implies that:} \\ \\ \alpha \beta = -1 \\ \\ \alpha + \beta = 1 \\ \\ \implies \alpha = ( 1-\beta) \\ \\ ( 1- \beta) \beta = -1 \\ \\ \implies \beta - \beta^2 = -1 \implies \beta - \beta^2 -1 = 0\\ \\ \beta = \dfrac{-(-1) \pm \sqrt{(-1)^2 -4(1)(-1)}}{2(1)}[/tex]
[tex]\beta = \dfrac{1\pm \sqrt{5}}{2} \\ \\ \beta = \dfrac{1 + \sqrt{5}}{2} \ \ and \ \ \alpha = \dfrac{1 - \sqrt{5}}{2}[/tex]
C)
[tex]\dfrac{1}{1-x-x^2}= \dfrac{A}{1-\alpha x}+ \dfrac{\beta}{1-\beta x} \\ \\ = \dfrac{A(1-\beta x) + B(1-\alpha x)}{(1-\alpha x) (1 - \beta x)} \\ \\ = \dfrac{(A+B)-(A\beta+B\alpha)x}{(1-\alpha x) (1-\beta x)}[/tex]
[tex]\text{It means:} \\ \\ A+B=1 \\ \\ B = (1-A) \\ \\ A\beta+ B \alpha =0 \\ \\ A\beta ( 1 -A) \alpha = 0 \\ \\ A( \beta - \alpha ) = -\alpha \\ \\ A = \dfrac{\alpha}{\alpha - \beta } \\ \\ \\ \\ B = 1 - \dfrac{\alpha }{\alpha - \beta} \implies \dfrac{\alpha - \beta - \alpha }{\alpha - \beta } \\ \\ =\dfrac{-\beta }{\alpha - \beta} \\ \\ \mathbf{B = \dfrac{\beta }{\beta - \alpha }}[/tex]
D)
[tex]\text{The formula for} a_n: \\ \\ a(x) = \dfrac{\alpha }{\alpha - \beta }\sum \limits ^{\infty}_{n=0} \alpha ^n x^n - \dfrac{\beta}{\beta - \alpha }\sum \limits ^{\infty}_{n=0} \beta x^n \\ \\ \implies \sum \limits ^{\infty}_{n =0} \dfrac{\alpha ^{n+1}- \beta ^{n+1}}{\alpha - \beta}x^n \\ \\ a_n = \dfrac{\alpha ^{n+1}- \beta ^{n+1}}{\alpha - \beta } \\ \\ \\ a_n = \dfrac{1}{\sqrt{5}} \Big (\Big( \dfrac{\sqrt{5}+1}{2}\Big)^{n+1}- \Big ( \dfrac{1-\sqrt{5}}{2}\Big) ^{n+1}\Big)[/tex]
A middle school took all of its 6th grade students on a field trip to see a ballet at a theater that has 4900 seats. The students filled 1078 of the seats in the theater. What percentage of the seats in the theater were filled by the 6th graders on the trip?
Consider the relationship 7r+2t=7. a. Write the relationship as a function r=f(t).
Answer: r = 2/7 -t
Step-by-step explanation: solve r from equation
7r = 2-7t l:7. r = 2/7 - t
The relationship as a function r=f(t) is [tex]\rm f(t)=\dfrac{7-2t}{7}\\\\[/tex].
What is the equation?In algebra, an equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value.
Consider the relationship;
7r+2t=7
Substitute r = f(t) in the equation
[tex]\rm 7r+2t=7\\\\7f(t)+2t=7\\\\ 7f(t)=7-2t\\\\f(t)=\dfrac{7-2t}{7}\\\\[/tex]
Hence, the relationship as a function r=f(t) is [tex]\rm f(t)=\dfrac{7-2t}{7}\\\\[/tex].
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Write an equation of the line in slope -intercept form ( due today!)
Answer: The answer would be y=1/3x-3
math, i can give you brainliset just help me with the two questions :)
Answer:
2. the interquartile range is 20
3. 80 is the median.
Step-by-step explanation:
have a nice day.
Please find the area of of the shaded polygon
Answer:
i think it might be 15... im not sure, im not good at math but im good with finding the answers to these type of problems
Step-by-step explanation:
hope this helps
Can you help me image is here which 2 Should I drag
Answer:
54, 63
Step-by-step explanation:
6 x 9 = 54
7 x 9 = 63
WILL MARL BRAINLEST
A full lake begins dropping at a constantrate. After 4 weeks it has dropped 3 feet.What is the unit rate of change in the lake’slevel compared to its full level?
A.0.75 feet per week
B.1.33 feet per week
C. −1.33 feet per week
D.−0.75 feet per week
1
Answer:
A is the answer
Step-by-step explanation:
A is the answer
Can you help me here is the inage
Answer:
150x=y
Step-by-step explanation:
2x150=300
4x150=600
6x150=900
8x150=1200
Someone PLEASE HELP ME WITH THIS IM IN A HURRY PLEASE
Answer:
A
Step-by-step explanation:
A
Convert the angle −4.5 radians to degrees, rounding to the nearest 10th
Answer:-257.8
Step-by-step explanation:
Help, please. The question is down below. I give brainliest. Please anyone help.
Answer:
25%
hope this helps
Find the slope and y-intercept for both problems
Answer:
1. 3/2
2. 4
Step-by-step explanation:
if one point is located at (-5,3) and another point is located at (7,-4). find the distance between the points. round to the nearest tenth.
Answer:
13.9
Step-by-step explanation:
Use the distance formula:
[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}\\\\d = \sqrt{(7 - (-5))^2 + (-4-3)^2}\\d = \sqrt{(12)^2 + (-7)^2}\\d = \sqrt{144 + 49}\\d = \sqrt{193}\\[/tex]
The square root of 193 rounded to the nearest tenth is 13.9.
HELP ME PLEASE
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Mr. Notar is shopping for a crib for his new baby. He finds a coupon for 15% off. If the crib originally costs $450, how much will he save with the coupon?
How can you use tables, graphs and equations to compare functions
Answer:
A graph is a chart or drawing that shows the relationship between changing things.
- They are a diagram displaying the relationship between numbers or amounts.
- A graph is one of the easiest ways to compare numbers.
- A graph or chart is used to present facts in visual form.
- They are a drawing that displays the relative sizes of numerical quantities.
- Graphs can be used to illustrate many types of data.
- They should be used to make facts clearer and more understandable.
- it shows trends over time.
- Forceful -- Emphasizes main point
- Convincing -- proves a point, see and hear
- Compact way to convey information
- More interesting than just talk or print
- Quick way for the audience to visualize what you are saying -- numbers trends up or down
- To bring facts to life
- To show and compare relationships
- To show and compare changes
- They are much easier to understand then a large
table of numbers
- without
the need to understand what each and every number in a table means.
They help you visually see the difference between two or more values.
Hope this helps!
What is the surface area?
5 yd
4 yd
5 yd
6 yd
6 yd
PLEASE HELP 100 points to who ever gives me the answer Compare the numbers using >, <, or =.
a) -2 3
b) |-12| |15|
c) 3 -4
d) |15| |-12|
e) 7 -11
Answer:
a) -2 < 3
b) |-12| < |15| (the absolute value of -12 is 12 so that's comparing 12 to 15)
c) 3 > -4
d) |15| > |-12|
e) 7 > -11
Please help!!!!!!!!!
Answer:D
Step-by-step explanation:
Solve for X
I would really appreciate any help! I need to finish this trigonometry course to graduate.
Find the errors and correct the following mathematical sentences.
Answer:
ix) (2a + 3)² = 4a² +12a + 9
x) a) x² + 7x + 12 = (-3)² + 7(-3) + 12 = 9 -21 + 12 = 0
b) x² - 5x + 6 = (-3)² - 5(-3) + 6 = 9 + 15 + 6 = 30
c) x² + 5x = (-3)² + 5(-3) = 9 -15 = -6
Please help due tomorrow please explain if can
Answer:
16) 9ft, C
17) 1440 cm^3, D
18) 314 m^2, B
Step-by-step explanation:
16) 18 / 6 = 27 / x
18x = 162
x = 9 (C)
17) 24 / 1,920 = 18 / x
24x = 34,560
x = 1440 (D)
18) Surface Area of a sphere = 4 * pi * radius^2 (4r^2)
SA = 4pi(5)^2
SA = 4pi25
SA = 100pi
SA = 100(~3.14)
SA = 314 (B)
i need help with b and jk
Answer:
b = 1
JK = 3
Step-by-step explanation:
5b - 2 = 9b - 6
4b = 4
b = 1
JK = 3
Solve -a + 7 = 2a - 8. a=5 D a= 15 a -3 = B a
SOLVE THESE PROBLEMS CORRECTLY TO BE MARKED BRAINLIEST!
1. Sophie wrote a two-digit number on the board. Max wrote the digit 3 to the right of it and got a number that is bigger than the Sophie's number by 372. What number did Sophie write on the board?
2. Some kids went on a bike ride. The ride was six hours long and the kids traveled 54 miles. For part of the ride they biked at a speed of 7 mph and for part of the ride they biked at a speed of 11 mph. For how long did they bike at the first speed, and for how long at the other speed?
3. I left Boston for Washington, DC, at 7:00 am. At first, my speed was 55 mph; then it was 75 mph. I traveled 630 miles in 10 hours. At what time did I begin to drive faster?
Thanks!
Answer: 1= 41
2= 3 hours biking at the other speed.
3= We began to drive at 75 mph at 1:00 p.m
Step-by-step explanation:
3= x/55 + (630 - x)/75 = 10
75x + 55 * (630 - x) = 41,250 (4,125 as Lowest common multiple)
75x + 34,650 - 55x = 41,250
20x = 41,250 - 34,650 (Subtracting 34,650 at both sides)
20x = 6,600
x = 6,600/20
x = 330 ⇒ 630 - x = 300
Time we drove at 55 mph = 330/55 = 6 hours
Time we drove at 75 mph = 300/75 = 4 hours
We began to drive at 75 mph at 1:00 p.m. (7 am + 6 hours)
hope this helps :)
WHAT ARE THESE TWO ILL GIVE BRAINLESS AND SHOW UR WORK
A car travels 46 feet in one sec . At this speed about how many feet will the car trave in 2 sec
Answer:
92 feet.
Step-by-step explanation:
pretty pls help with this questionn pls i need helpp
Plzz help mee I am timed
Answer:
A or D
Step-by-step explanation:
I cant totally remember, but a and d make sense. sorry, I learned about this in 6th
