# 11.7 Probability  (Page 9/18)

 Page 9 / 18

How many subsets does the set have?

${2}^{50}=1.13×{10}^{15}$

A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa?

How many distinct ways can the word DEADWOOD be arranged?

$\frac{8!}{3!2!}=3360$

How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?

## Binomial Theorem

Evaluate the binomial coefficient $\text{\hspace{0.17em}}\left(\begin{array}{c}23\\ 8\end{array}\right).$

$490\text{,}314$

Use the Binomial Theorem to expand ${\left(3x+\frac{1}{2}y\right)}^{6}.$

Use the Binomial Theorem to write the first three terms of ${\left(2a+b\right)}^{17}.$

$131\text{,}072{a}^{17}\text{+}1\text{,}114\text{,}112{a}^{16}b\text{+}4\text{,}456\text{,}448{a}^{15}{b}^{2}$

Find the fourth term of ${\left(3{a}^{2}-2b\right)}^{11}$ without fully expanding the binomial.

## Probability

For the following exercises, assume two die are rolled.

Construct a table showing the sample space.

 1 2 3 4 5 6 1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

What is the probability that a roll includes a $2?$

What is the probability of rolling a pair?

$\frac{1}{6}$

What is the probability that a roll includes a 2 or results in a pair?

What is the probability that a roll doesn’t include a 2 or result in a pair?

$\frac{5}{9}$

What is the probability of rolling a 5 or a 6?

What is the probability that a roll includes neither a 5 nor a 6?

$\frac{4}{9}$

For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

What is the percent chance that all the children attending the party prefer soda?

What is the percent chance that at least one of the children attending the party prefers milk?

$1-\frac{C\left(350,8\right)}{C\left(500,8\right)}\approx 94.4%$

What is the percent chance that exactly 3 of the children attending the party prefer soda?

What is the percent chance that exactly 3 of the children attending the party prefer milk?

$\frac{C\left(150,3\right)C\left(350,5\right)}{C\left(500,8\right)}\approx 25.6%$

## Practice test

Write the first four terms of the sequence defined by the recursive formula

$-14,-6,-2,0$

Write the first four terms of the sequence defined by the explicit formula ${a}_{n}=\frac{{n}^{2}–n–1}{n!}.$

Is the sequence arithmetic? If so find the common difference.

The sequence is arithmetic. The common difference is $d=0.9.$

An arithmetic sequence has the first term ${a}_{1}=-4$ and common difference $d=–\frac{4}{3}.$ What is the 6 th term?

Write a recursive formula for the arithmetic sequence and then find the 22 nd term.

Write an explicit formula for the arithmetic sequence and then find the 32 nd term.

Is the sequence $\text{\hspace{0.17em}}-2\text{,}-1\text{,}-\frac{1}{2}\text{,}-\frac{1}{4}\text{,}\dots$ geometric? If so find the common ratio. If not, explain why.

The sequence is geometric. The common ratio is $r=\frac{1}{2}.$

What is the 11 th term of the geometric sequence $\text{\hspace{0.17em}}-1.5,-3,-6,-12,\dots ?$

Write a recursive formula for the geometric sequence

Write an explicit formula for the geometric sequence

Use summation notation to write the sum of terms $\text{\hspace{0.17em}}3{k}^{2}-\frac{5}{6}k\text{\hspace{0.17em}}$ from $\text{\hspace{0.17em}}k=-3\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}k=15.$

$\sum _{k=-3}^{15}\left(3{k}^{2}-\frac{5}{6}k\right)$

A community baseball stadium has 10 seats in the first row, 13 seats in the second row, 16 seats in the third row, and so on. There are 56 rows in all. What is the seating capacity of the stadium?

Use the formula for the sum of the first $n$ terms of a geometric series to find $\sum _{k=1}^{7}-0.2\cdot {\left(-5\right)}^{k-1}.$

${S}_{7}=-2604.2$

Find the sum of the infinite geometric series $\sum _{k=1}^{\infty }\frac{1}{3}\cdot {\left(-\frac{1}{5}\right)}^{k-1}.$

Rachael deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned? Total in account: $\text{}140,355.75;$ Interest earned: $\text{}14,355.75$ In a competition of 50 professional ballroom dancers, 22 compete in the fox-trot competition, 18 compete in the tango competition, and 6 compete in both the fox-trot and tango competitions. How many dancers compete in the fox-trot or tango competitions? A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have? $5×3×2×3×2=180$ To allocate annual bonuses, a manager must choose his top four employees and rank them first to fourth. In how many ways can he create the “Top-Four” list out of the 32 employees? A rock group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if have 15 songs to pick from? $C\left(15,3\right)=455$ A self-serve frozen yogurt shop has 8 candy toppings and 4 fruit toppings to choose from. How many ways are there to top a frozen yogurt? How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E? $\frac{10!}{2!3!2!}=151\text{,}200$ Use the Binomial Theorem to expand ${\left(\frac{3}{2}x-\frac{1}{2}y\right)}^{5}.$ Find the seventh term of ${\left({x}^{2}-\frac{1}{2}\right)}^{13}$ without fully expanding the binomial. $\frac{429{x}^{14}}{16}$ For the following exercises, use the spinner in [link] . Construct a probability model showing each possible outcome and its associated probability. (Use the first letter for colors.) What is the probability of landing on an odd number? $\frac{4}{7}$ What is the probability of landing on blue? What is the probability of landing on blue or an odd number? $\frac{5}{7}$ What is the probability of landing on anything other than blue or an odd number? A bowl of candy holds 16 peppermint, 14 butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.) $\frac{C\left(14,3\right)C\left(26,4\right)}{C\left(40,7\right)}\approx 29.2%$ #### Questions & Answers I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once Carlos Reply How can you tell what type of parent function a graph is ? Mary Reply generally by how the graph looks and understanding what the base parent functions look like and perform on a graph William if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero William y=x will obviously be a straight line with a zero slope William y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis William y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer. Aaron yes, correction on my end, I meant slope of 1 instead of slope of 0 William what is f(x)= Karim Reply I don't understand Joe Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain." Thomas Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-) Thomas GREAT ANSWER THOUGH!!! Darius Thanks. Thomas Â Thomas It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â Thomas Now it shows, go figure? Thomas what is this? unknown Reply i do not understand anything unknown lol...it gets better Darius I've been struggling so much through all of this. my final is in four weeks 😭 Tiffany this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts Darius thank you I have heard of him. I should check him out. Tiffany is there any question in particular? Joe I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously. Tiffany Sure, are you in high school or college? Darius Hi, apologies for the delayed response. I'm in college. Tiffany how to solve polynomial using a calculator Ef Reply So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right? KARMEL Reply The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26 Rima Reply The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer? Rima I done know Joe What kind of answer is that😑? Rima I had just woken up when i got this message Joe Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that Rima i have a question. Abdul how do you find the real and complex roots of a polynomial? Abdul @abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up Nare This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1 Abdul @Nare please let me know if you can solve it. Abdul I have a question juweeriya hello guys I'm new here? will you happy with me mustapha The average annual population increase of a pack of wolves is 25. Brittany Reply how do you find the period of a sine graph Imani Reply Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period Am if not then how would I find it from a graph Imani by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates. Am you could also do it with two consecutive minimum points or x-intercepts Am I will try that thank u Imani Case of Equilateral Hyperbola Jhon Reply ok Zander ok Shella f(x)=4x+2, find f(3) Benetta f(3)=4(3)+2 f(3)=14 lamoussa 14 Vedant pre calc teacher: "Plug in Plug in...smell's good" f(x)=14 Devante 8x=40 Chris Explain why log a x is not defined for a < 0 Baptiste Reply the sum of any two linear polynomial is what Esther Reply divide simplify each answer 3/2÷5/4 Momo Reply divide simplify each answer 25/3÷5/12 Momo how can are find the domain and range of a relations austin Reply the range is twice of the natural number which is the domain Morolake A cell phone company offers two plans for minutes. Plan A:$15 per month and $2 for every 300 texts. Plan B:$25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money? Diddy Reply 6000 Robert more than 6000 Robert For Plan A to reach$27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert