# 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.)

$\frac{C\left(14,3\right)C\left(26,4\right)}{C\left(40,7\right)}\approx 29.2%$

can you not take the square root of a negative number
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
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
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
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)=
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
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