13.7 Probability  (Page 7/18)

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What is the percent chance that a player selects exactly 3 winning numbers?

$\text{\hspace{0.17em}}\frac{C\left(20,3\right)C\left(60,17\right)}{C\left(80,20\right)}\approx 12.49%\text{\hspace{0.17em}}$

What is the percent chance that a player selects exactly 4 winning numbers?

What is the percent chance that a player selects all 5 winning numbers?

$\text{\hspace{0.17em}}\frac{C\left(20,5\right)C\left(60,15\right)}{C\left(80,20\right)}\approx 23.33%\text{\hspace{0.17em}}$

What is the percent chance of winning?

How much less is a player’s chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers?

$20.50+23.33-12.49=31.34%$

Real-world applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). United States Census Bureau. http://www.census.gov

If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

$\frac{C\left(40000000,1\right)C\left(277000000,4\right)}{C\left(317000000,5\right)}=36.78%$

If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

$\frac{C\left(40000000,4\right)C\left(277000000,1\right)}{C\left(317000000,5\right)}=0.11%$

It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true?

Sequences and Their Notation

Write the first four terms of the sequence defined by the recursive formula $\text{\hspace{0.17em}}{a}_{1}=2,\text{\hspace{0.17em}}{a}_{n}={a}_{n-1}+n.$

$2,4,7,11$

Evaluate $\text{\hspace{0.17em}}\frac{6!}{\left(5-3\right)!3!}.$

Write the first four terms of the sequence defined by the explicit formula $\text{\hspace{0.17em}}{a}_{n}={10}^{n}+3.$

$13,103,1003,10003$

Write the first four terms of the sequence defined by the explicit formula $\text{\hspace{0.17em}}{a}_{n}=\frac{n!}{n\left(n+1\right)}.$

Arithmetic Sequences

Is the sequence $\text{\hspace{0.17em}}\frac{4}{7},\frac{47}{21},\frac{82}{21},\frac{39}{7},\text{\hspace{0.17em}}...$ arithmetic? If so, find the common difference.

The sequence is arithmetic. The common difference is $\text{\hspace{0.17em}}d=\frac{5}{3}.$

Is the sequence $\text{\hspace{0.17em}}2,4,8,16,\text{\hspace{0.17em}}...\text{\hspace{0.17em}}$ arithmetic? If so, find the common difference.

An arithmetic sequence has the first term $\text{\hspace{0.17em}}{a}_{1}=18\text{\hspace{0.17em}}$ and common difference $\text{\hspace{0.17em}}d=-8.\text{\hspace{0.17em}}$ What are the first five terms?

$18,10,2,-6,-14$

An arithmetic sequence has terms ${a}_{3}=11.7$ and ${a}_{8}=-14.6.$ What is the first term?

Write a recursive formula for the arithmetic sequence $-20\text{,}-10,0\text{,}10\text{,…}$

Write a recursive formula for the arithmetic sequence and then find the 31 st term.

Write an explicit formula for the arithmetic sequence

${a}_{n}=\frac{1}{3}n+\frac{13}{24}$

How many terms are in the finite arithmetic sequence $\text{\hspace{0.17em}}12,20,28,\dots ,172?$

Geometric Sequences

Find the common ratio for the geometric sequence

$r=2$

Is the sequence geometric? If so find the common ratio. If not, explain why.

A geometric sequence has terms $\text{\hspace{0.17em}}{a}_{7}=16\text{,}384\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}_{9}=262\text{,}144\text{\hspace{0.17em}.}$ What are the first five terms?

A geometric sequence has the first term $\text{\hspace{0.17em}}{a}_{1}\text{=}-3\text{\hspace{0.17em}}$ and common ratio $\text{\hspace{0.17em}}r=\frac{1}{2}.\text{\hspace{0.17em}}$ What is the 8 th term?

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
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Prove it
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Eseka
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