Deck 17: Probability Web

Question 1

A computer manufacturer offers a computer system with three different disk drives, two different monitors, and five different keyboards. How many different computer systems could a consumer purchase from this manufacturer?

A)16
B)24
C)30
D)20
E)60

Answer 1

30

Question 2

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing three green, five yellow, and four red marbles such that both marbles are yellow.

A) $\frac { 5 } { 33 }$
B) $\frac { 25 } { 144 }$
C) $\frac { 25 } { 132 }$
D) $\frac { 2 } { 5 }$
E) $\frac { 5 } { 7 }$

Answer 2

$\frac { 5 } { 33 }$

Question 3

Find the number of distinguishable permutations of the group of letters. $E , S , T , I , M , A , T , E$

A) $10,080$
B) $8$
C) $20,160$
D) $40,320$
E) $3360$

Answer 3

$10,080$

Question 4

In how many ways can a 9-question true-false exam be answered? (Assume that no questions are omitted.)

A)512
B)131,072
C)8192
D)524,288
E)4096

Question 5

Determine the number of ways a computer can randomly generate an integer divisible by 4 from 1 through 15.

A)6
B)3
C)11
D)7
E)14

Question 6

A computer randomly generates an integer from 1 through 50. Find the probability of the event that a multiple of 3 is generated.

A) $\frac { 7 } { 50 }$
B) $\frac { 8 } { 25 }$
C) $\frac { 3 } { 25 }$
D) $\frac { 1 } { 5 }$
E) $\frac { 1 } { 10 }$

Question 7

A state lottery game requires a person to select ten different numbers from thirty-three numbers. The order of the selection is not important. In how many ways can this be done?

A) $286$
B) $1,144,066$
C) $92,561,040$
D) $330$
E) $2 , \mathrm { e } 10$

Question 8

Eighteen students are selected as semifinalists for a literary award. Of the eighteen students, eight finalists will be selected. In how many ways can eight finalists be selected from the eighteen students?

A)6435
B)203,490
C)24,310
D)43,758
E)125,970

Question 9

In a certain state, each automobile license plate number consists of two letters followed by a four-digit number. To avoid confusion between "O" and "zero" and "I" and "one", the letters "O" and "I" are not used. How many distinct license plate numbers can be formed?

A)57,600,000
B)5,760,000
C)13,824,000
D)138,240,000
E)331,776,000

Question 10

Evaluate: 7P3

A)35
B)840
C)210
D)21
E)undefined

Question 11

A combination lock will open when the right choice of three numbers (from 1 to 32) is selected. How many different lock combinations are possible?

A)98,304
B)1024
C)32
D)96
E)32,768

Question 12

Evaluate the expression. ${ } _ { 5 } C _ { 4 }$

A)5
B)20
C)625
D)1024
E)9

Question 13

There are 11 patients in Dr. Ziglar's waiting room. Dr. Ziglar can see 5 patients before lunch. In how many different orders can Dr. Ziglar see 5 of the patients before lunch?

A)462
B)332,640
C)55
D)5
E)55,440

Question 14

Thirteen students, of whom three are seniors, are selected as semifinalists for a literary award. Of the thirteen students, nine finalists will be selected. In how many ways can the nine finalists contain two seniors?

A) $715$
B) $120$
C) $2145$
D) $360$
E) $55$

Question 15

At a high school cafeteria, diners can choose one vegetable from a choice of 2 vegetables, one meat from a choice of 3 meats, one serving of bread from among 4 breads, and a dessert from among 4 desserts. How many meal configurations are possible?

A)13
B)96
C)4
D)24
E)48

Question 16

Decide which of the scenarios below should be counted using permutations or combinations. Scenario I:
Number of ways 12 movies can be ordered to play on television.
Scenario II:
Number of ways three different roles can be filled by 11 people auditioning for a play.
Scenario III:
Number of different two-topping pizzas that can be made from an assortment of 10 different toppings.

A)Scenario III is a combination and scenarios I and II are permutations.
B)All scenarios are combinations.
C)Scenario II is a combination and scenarios I and III are permutations.
D)Scenarios I and II are combinations and scenario III is a permutation.
E)All scenarios are permutations.

Question 17

Seven cards are chosen at random from a standard deck of playing cards. In how many ways can the cards be chosen if all seven cards are spades.

A) $1716$
B) $33,446,140$
C) $4$
D) $5040$
E) $26,545$

Question 18

A card is selected from a standard deck of 52 cards. Find the probability of getting a card that is less than 4 (aces are low).

A) $\frac { 11 } { 13 }$
B) $\frac { 4 } { 13 }$
C) $\frac { 5 } { 13 }$
D) $\frac { 3 } { 13 }$
E) $\frac { 2 } { 13 }$

Question 19

A coin is tossed three times. Find the probability of getting at least two heads.

A) $\frac { 1 } { 2 }$
B) $\frac { 7 } { 8 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 8 }$
E) $\frac { 3 } { 4 }$

Question 20

Two six-sided dice are tossed. Find the probability that the sum is odd.

A) $\frac { 5 } { 6 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 5 } { 18 }$
E) $\frac { 23 } { 36 }$

Question 21

Find $P ( x \leq 3 )$ given the probability distribution. $$\begin{array} { l l l l l l l } x & 0 & 1 & 2 & 3 & 4 & 5 \\P ( x ) & 0.045 & 0.181 & 0.248 & 0.328 & 0.154 & 0.044\end{array}$$

A)0.328
B)0.474
C)0.526
D)0.802
E)0.198

Question 22

Estimate the variance for the following probability distribution to two decimal places. $$\begin{array} { c | c c c c } x & 0 & 1 & 2 & 3 \\\hline \mathrm { P } ( x ) & 1 / 10 & 1 / 20 & 1 / 20 & 4 / 5\end{array}$$

A) $2.55$
B) $0.95$
C) $0.97$
D) $0.90$
E)1.00

Question 23

Determine whether the table represents a probability distribution. $$\begin{array} { | l | l | l | l | l | } \hline x & 0 & 1 & 2 & 3 \\\hline P ( x ) & 0.10 & 0.45 & 0.30 & 0.15 \\\hline\end{array}$$

A)yes
B)no

Question 24

Assume that the probability of the birth of a child of a particular gender is 50%. In a family with seven children, what is the probability that there is at least one girl?

A) $\frac { 127 } { 128 }$
B) $\frac { 59 } { 128 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 3 } { 32 }$
E) $\frac { 43 } { 128 }$

Question 25

A sales representative makes a sale at approximately one-third of the businesses he calls on. On a given day, he goes to four businesses. What is the probability that he will make a sale at all four businesses?

A) $\frac { 8 } { 27 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 12 }$
D) $\frac { 1 } { 81 }$
E) $\frac { 16 } { 81 }$

Question 26

A coin is tossed three times. Describe the event A that at least two heads occur.

A) $\{ \mathrm { HHH } , \mathrm { HTT } , \mathrm { HTH } , \mathrm { TTH } \}$
B) $\{ \mathrm { TTT } , \mathrm { HHT } , \mathrm { HTH } , \mathrm { THH } \}$
C) $\{ \mathrm { HHH } , \mathrm { HHT } , \mathrm { HTH } , \mathrm { THH } \}$
D) $\{ \mathrm { HHHH } , \mathrm { HHHT } , \mathrm { HTHH } , \mathrm { TTHH } \}$
E) $\{ \mathrm { HH } , \mathrm { HT } , \mathrm { TT } \}$

Question 27

You are given the probability that an event will not happen. Find the probability that the event will happen. $P \left( E ^ { \prime } \right) = \frac { 3 } { 31 }$

A) $\frac { 3 } { 31 }$
B)0
C)1
D) $\frac { 28 } { 31 }$
E) $\frac { 14 } { 31 }$

Question 28

Sketch a graph of the probability distribution. $$\begin{array} { l c c c c c } x & 0 & 1 & 2 & 3 & 4 \\P ( x ) & \frac { 1 } { 25 } & \frac { 6 } { 25 } & \frac { 8 } { 25 } & \frac { 3 } { 25 } & \frac { 7 } { 25 }\end{array}$$

A)
B)
C)
D)
E)

Question 29

A card is chosen at random from two 52-card decks of playing cards" . What is the probability that the card will be black and a face card? A face card is a king, a queen, or a jack.

A) $\frac { 1 } { 13 }$
B) $\frac { 3 } { 52 }$
C) $\frac { 2 } { 13 }$
D) $\frac { 3 } { 26 }$
E) $\frac { 5 } { 26 }$

Question 30

You are given the probability that an event will happen. Find the probability that the event will not happen. P(E) = 0.29

A)0.29
B)0.71
C)0.355
D)0
E)1

Question 31

Find the standard deviation $\sigma$ for the following probability distribution. Round your answer to three decimal places. $$\begin{array} { l l l l c c } x & 1 & 2 & 3 & 4 & 5 \\P ( x ) & \frac { 2 } { 5 } & \frac { 1 } { 10 } & \frac { 1 } { 10 } & \frac { 1 } { 10 } & \frac { 3 } { 10 }\end{array}$$

A)2.800
B)1.720
C)8.762
D)2.960
E)1.673

Question 32

A quality control inspector receives a shipment of 30 computer monitors. From the 30 monitors, the inspector randomly chooses 8 for inspection. If the probability of a monitor being defective is 0.07, what is the probability that at least one of the monitors chosen by the inspector is defective? Round to the nearest hundredth.

A)0.44
B)0.52
C)0.51
D)0.47
E)0.45

Question 33

In a survey, Americans were asked how well informed they are about new scientific discoveries. The results are shown in the pie graph below. If two people from the survey are chosen at random, what is the probability that neither person feels very informed about scientific discoveries? Round to the nearest ten-thousandth.

A)0.3864
B)0.7921
C)0.8464
D)0.6853
E)0.7084

Question 34

Find $P ( x \leq 3 )$ given the probability distribution. $\begin{array} { l l l l l } x & 0 & 1 & 2 & 3 \\P ( x ) & 0.020 & 0.186 & 0.450 & 0.344\end{array}$

A)0.344
B)0.656
C)0.794
D)1.000
E)0.250

Question 35

One card is randomly drawn from a standard deck of playing cards where aces are not considered numbered cards and are the highest card in the suit. The card is replaced and another card is drawn. What is the probability that both cards drawn are numbered cards greater than two?

A) $\frac { 1 } { 169 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 49 } { 169 }$
D) $\frac { 64 } { 169 }$
E) $\frac { 25 } { 169 }$

Question 36

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing three green, six yellow, and four red marbles such that the marbles are different colors.

A) $\frac { 9 } { 26 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 9 } { 13 }$
D) $\frac { 2 } { 13 }$
E) $\frac { 2 } { 3 }$

Question 37

A shipment of nine calculators contains five defective calculators. Three calculators are chosen from the shipment. Find the probability that exactly three are defective.

A) $\frac { 3 } { 84 }$
B) $\frac { 5 } { 42 }$
C) $\frac { 23 } { 3 }$
D) $\frac { 1 } { 42 }$
E) $\frac { 5 } { 84 }$

Question 38

Find the expected value $E ( x )$ for the following probability distribution. Round your answer to three decimal places. $x$ $-$ 5,000 $-$ 2,500 300 $P ( x )$ 0.012
0.052
0.936

A)9.529
B)837.255
C)8,244.640
D)700,995.360
E)90.800

Question 39

Two people are asked their opinions on a political issue. They can answer "Opposed" (O) or "Undecided" (U). Find the sample space S.

A) $\{ \mathrm { OO } , \mathrm { OU } , \mathrm { UO } , \mathrm { UUU } \}$
B) $\{ \mathrm { OO } , \mathrm { OU } , \mathrm { UUO } , \mathrm { UU } \}$
C) $\{ \mathrm { OOO } , \mathrm { OU } , \mathrm { UOU } , \mathrm { UU } \}$
D) $\{ \mathrm { OOO } , \text { OUU, UOU, UUU } \}$
E) $\{ O O , \mathrm { OU } , \mathrm { UO } , \mathrm { UU } \}$

Question 40

A biology instructor gives her class a list of eight study problems, from which she will select five to be answered on an exam. A student knows how to solve six of the problems. Find the probability that the student will be able to answer all five questions on the exam.

A) $\frac { 23 } { 28 }$
B) $\frac { 3 } { 28 }$
C) $\frac { 11 } { 14 }$
D) $\frac { 1 } { 28 }$
E) $\frac { 27 } { 28 }$

Question 41

A baseball fan examined the record of a favorite baseball player's performance during his last 50 games. The numbers of games in which the player had zero, one, two, three, and four hits are recorded in the table shown below. Find the standard deviation $\sigma$ . Round your answer to two decimal places. $$\begin{array} { l l l l l l } \text { Number of hits } & 0 & 1 & 2 & 3 & 4 \\\text { Frequency } & 14 & 24 & 7 & 3 & 2\end{array}$$

A)1.10
B)1.00
C)1.02
D)1.01
E)1.05

Question 42

Find the mean, variance, and standard deviation of the uniform distribution $f ( x ) = \frac { 1 } { 18 }$ over the interval [0, 18] without using integration.

A) $\begin{array} { l l } \text { expected value (mean) : } & 9.000 \\\text { variance: } & 27.000 \\\text { standard deviation: } & 5.196\end{array}$
B) $\begin{array} { l l } \text { expected value (mean) : } & 27.000 \\\text { variance: } & 9.000 \\\text { standard deviation: } & 5.196\end{array}$
C) $\begin{array} { l l } \text { expected value (mean) : } & 27.000 \\\text { variance: } & 5.196 \\\text { standard deviation: } & 9.000\end{array}$
D) $\begin{array} { l l } \text { expected value (mean) } : & 5.196 \\\text { variance: } & 9.000 \\\text { standard deviation: } & 27.000\end{array}$
E) $\begin{array} { l l } \text { expected value (mean) } & 9.000 \\\text { variance: } & 5.196 \\\text { standard deviation: } & 27.000\end{array}$

Question 43

Find the value of the constant a that makes the given function a probability density function on the stated interval. $f ( x ) = ax^ { 2 } ( 10 - x ) \text { on } [ 0,1 ]$

A) $\frac { 6 } { 37 }$
B) $\frac { 37 } { 12 }$
C) $\frac { 12 } { 37 }$
D) $\frac { 37 } { 6 }$
E)1

Question 44

For the probability density function $f ( x ) = \frac { x } { 128 }$ on the interval $[ 0,16 ]$ , find the probability that $x \geq 6$ . Round your answer to the nearest hundredth.

A)0.1
B)0.63
C)0.86
D)0.08
E)0.06

Question 45

The daily demand for gasoline (in millions of gallons) in a city is described by the probability density function $f ( x ) = \frac { 1 } { 5 } - \frac { 1 } { 50 } x$ over the interval $[ 0,10 ]$ . Find the probability that the daily demand for gasoline will be at least 3 million gallons.

A)0.640
B)0.360
C)0.423
D)0.562
E)0.490

Question 46

Sketch the graph of the probability density function $f ( x ) = \frac { 1 } { 3 }$ over the interval $[ 0,3 ]$ .

A)
B)
C)
D)
E)

Question 47

For the probability density function $f ( x ) = \frac { 7 } { 6 ( x + 1 ) ^ { 2 } }$ on the interval $[ 0,6 ]$ find the probability that $x \leq 4$ . Round your answer to the nearest hundredth.

A)0.05
B)0.33
C)0.93
D)0.67
E)0.02

Question 48

For the given probability density function, find $\operatorname { Var } ( X )$ $f ( x ) = \frac { 3 } { 8 } x ^ { 2 } \text { on } [ 0,2 ]$

A)2.000
B)1.150
C)1.500
D)0.150
E)0.800

Question 49

The time t (in hours) required for a new employee to successfully learn to operate a machine in a manufacturing process is described by the probability density function $f ( t ) = \frac { 5 } { 324 } t \sqrt { 9 - t }$ over the interval $[ 0,9 ]$ . Find the probability that a new employee will learn to operate the machine in more than 3 hours but less than 7 hours.

A)0.6634
B)0.3052
C)0.5895
D)0.2733
E)0.4632

Question 50

A meteorologist predicts that the amount of rainfall (in inches) expected for a certain coastal community during a hurricane has the probability density function $f ( x ) = \frac { \pi } { 30 } \sin \frac { \pi x } { 15 },$ $0 \leq x \leq 15$ . Find and interpret the probability $P ( 0 \leq x \leq 5 )$ .

A)17% probability of receiving up to 5 inches of rain
B)25% probability of receiving up to 5 inches of rain
C)30% probability of receiving up to 5 inches of rain
D)75% probability of receiving up to 5 inches of rain
E)83% probability of receiving up to 5 inches of rain

Question 51

For the probability density function $f ( t ) = \frac { 1 } { 2 } e ^ { - \frac { t } { 2 } }$ on the interval $[ 0 , \infty )$ , find the probability that $t \geq 3$ . Round your answer to the nearest thousandth.

A)0.112
B)0.388
C)0.183
D)0.333
E)0.223

Question 52

Find the median of the exponential probability density function $f ( t ) = \frac { 4 } { 5 } e ^ { - \frac { 4 t } { 5 } }$ over the interval $[ 0 , \infty )$ .

A)0.555
B)0.693
C)0.866
D)0.485
E)0.400

Question 53

Sketch the graph of the following probability density function and locate the mean on the graph. $f ( t ) = \frac { t } { 18 } , [ 0,6 ]$

A) mean :1.5
B) mean :2.0
C) mean :4.0
D) mean :3.0
E) mean :4.0

Question 54

A publishing company introduces a new weekly magazine that sells for $4.95 on the newsstand. The marketing group of the company estimates that sales x (in thousands) will be approximated by the following probability function. Find the expected revenue. Round your answer to the nearest dollar. $$\begin{array} { l l l l l l } x & 10 & 15 & 20 & 30 & 40 \\P ( x ) & 0.23 & 0.28 & 0.23 & 0.14 & 0.12\end{array}$$

A) $\$$ 99,495
B) $\$$ 99
C) $\$$ 455
D) $\$$ 20,100
E) $\$$ 455,351

Question 55

Find the constant k such that the function $f ( x ) = k x$ is a probability density function over the interval $[ 3,8 ]$ .

A)5
B) $\frac { 2 } { 55 }$
C) $\frac { 2 } { 5 }$
D) $\frac { 55 } { 2 }$
E) $\frac { 5 } { 2 }$

Question 56

Find the constant $k$ such that the function $f ( x ) = k e ^ { - \frac { 9 } { 7 } x }$ is a probability density function over the interval $[ 0 , \infty ]$ .

A)1
B) $\frac { 7 } { 9 }$
C) $- \frac { 7 } { 9 }$
D) $- \frac { 9 } { 7 }$
E) $\frac { 9 } { 7 }$

Question 57

Buses arrive and depart from a college every 25 minutes. The probability density function for the waiting time t (in minutes) for a person arriving at the bus stop is $f ( t ) = \frac { 1 } { 25 }$ on the interval $[ 0,25 ]$ . Find the probability that the person will wait no longer than 10 minutes.

A) $\frac { 1 } { 25 }$
B) $\frac { 1 } { 10 }$
C) $\frac { 2 } { 5 }$
D) $\frac { 3 } { 5 }$
E) $\frac { 1 } { 250 }$

Question 58

Sketch the graph of the following probability density function and locate the mean on the graph. $f ( x ) = \frac { 5 } { 2 } x ^ { 3 / 2 } , [ 0,1 ]$

A) mean :0.714
B) mean :1.600
C) mean :2.012
D) mean :0.555
E) mean :1.848

Question 59

If x is the net gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose over the long run. A service organization is selling $2 raffle tickets as part of a fundraising program. The first prize is a boat valued at $2950, and the second prize is a camping tent valued at $400. In addition to the first and second prizes, there are twenty-four $23 gift certificates to be awarded. The number of tickets sold is 3000. Find the expected net gain to the player for one play of the game. Round your answer to the nearest cent.

A) $\$$ 1122.33
B) $\$$ 1.40
C) $\$$ 0.70
D)-$0.7
E)-$1122.33

Question 60

An insurance company needs to determine the annual premium required to break even on fire protection policies with a face value of $\$$ 80,000. If x is the claim size on these policies and the analysis is restricted to the losses $\$$ 30,000, $\$$ 60,000, and $\$$ 80,000, then the probability distribution of x is as shown in the table. What premium should customers be charged for the company to break even? Round your answer to the nearest dollar. $$\begin{array} { l l l l l } x & 0 & 30,000 & 60,000 & 80,000 \\P ( x ) & 0.9940 & 0.0032 & 0.0010 & 0.0018\end{array}$$

A) $\$$ 298
B) $\$$ 17
C) $\$$ 42,500
D) $\$$ 300
E) $\$$ 50,000

Question 61

Let $x$ be a random variable that is normally distributed with the given mean $\mu = 48$ and standard deviation $\sigma = 8$ . Find the probability $P ( x > 55 )$ using a symbolic integration utility. Round your answer to four decimal places.

A)0.2525
B)0.1908
C)0.3085
D)0.0763
E)0.1223

Question 62

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 11 } \left( 8 n - 4 n ^ { 2 } \right)$

A)-1100
B)-4488
C)-396
D)264
E)-1496

Question 63

Let $x$ be a random variable that is normally distributed with the given mean $\mu = 50$ and standard deviation $\sigma = 10$ . Find the probability $P ( 30 < x < 55 )$ using a symbolic integration utility.

A)0.7970
B)0.5889
C)0.6687
D)0.5359
E)0.6302

Question 64

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 7 , a _ { 2 } = 5 , a _ { 4 } = 11$

A) $a _ { n } = n ^ { 2 } + 7$
B) $a _ { n } = n ^ { 2 } - 3 n + 7$
C) $a _ { n } = n ^ { 2 } - 7 n + 7$
D) $a _ { n } = n ^ { 2 } - 7 n + 3$
E) $a _ { n } = n ^ { 2 } + n + 7$

Question 65

Evaluate the binomial coefficient. $\left( \begin{array} { l } 9 \\6\end{array} \right)$

A) $504$
B) $84$
C) $362,880$
D) $6$
E) $720$

Question 66

The arrival time t of a bus at a bus stop is uniformly distributed between $08 : 00$ A.M. and $08 : 08$ A.M. What is the probability that you will miss the bus if you arrive at the bus stop at $08 : 02$ A.M.? Round your answer to two decimal places.

A)0.25
B)0.67
C)0.36
D)0.33
E)0.57

Question 67

Use mathematical induction to prove the formula for every positive integer n. Show all your work.

Question 68

Find Pk+1 for the given Pk. $P _ { k } = \frac { 6 } { k ( k + 1 ) }$

A) $P _ { k + 1 } = \frac { 6 } { k ( k + 1 ) } + 1$
B) $P _ { k + 1 } = \frac { 6 } { k ( k + 1 ) } + \frac { 6 } { ( k + 1 ) ( k + 2 ) }$
C) $P _ { k + 1 } = \frac { 6 } { ( k + 1 ) ( k + 2 ) }$
D) $P _ { k + 1 } = \frac { 6 } { k ( k + 2 ) }$
E) $P _ { k + 1 } = \frac { 36 } { ( k + 1 ) ( k + 2 ) }$

Question 69

The daily demand $x$ for a certain product (in hundreds of pounds) is a random variable with the probability density function $f ( x ) = \frac { 2 } { 9 } x ( 3 - x )$ over the interval $[ 0,3 ]$ . Find the probability that $x$ is within one standard deviation of the mean. Express your answer as a percent.

A)96.44 $\%$
B)49.28 $\%$
C)62.61 $\%$
D)13.41 $\%$
E)72.77 $\%$

Question 70

Find Pk+1 for the given Pk. $P _ { k } = \frac { k ^ { 2 } } { 6 } ( 5 k + 5 )$

A) $P _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 6 } ( 5 k + 10 )$
B) $P _ { k + 1 } = \frac { k ^ { 2 } } { 6 } ( 5 k + 5 ) + 1$
C) $P _ { k + 1 } = \frac { k ^ { 2 } } { 6 } ( 5 k + 10 )$
D) $P _ { k + 1 } = \frac { k ^ { 2 } + 1 } { 6 } ( 5 k + 10 )$
E) $P _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 6 } ( 5 k + 5 )$

Question 71

If $X$ is an exponential random variable with the probability density function $f ( x ) = 23 e ^ { - 23 x } \text { on } [ 0 , \infty ),$ find $E ( X )$

A) $46$
B) $23$
C) $\frac { 1 } { 23 }$
D) $\frac { 2 } { 23 }$
E)1

Question 72

Find the mean, variance, and standard deviation of the normal density function $f ( x ) = \frac { 1 } { 2 \sqrt { 2 \pi } } e ^ { - ( x -- 1 ) ^ { 2 }/8}$ over $( - \infty , \infty )$ . Do not use integration.

A) $\begin{array} { l l } \text { expected value(mean): } & - 1 \\\text { variance: } & 4 \\\text { standard deviation: } & 2\end{array}$
B) $\begin{array} { l l } \text { expected value (mean) } ) : & 4 \\\text { variance: } & - 1 \\\text { standard dev iation: } & 2\end{array}$
C) $\begin{array} { l l } \text { expected value(mean } ) : & 4 \\\text { variance: } & 2 \\\text { standard dev iation: } & - 1\end{array}$
D) $\begin{array} { l l } \text { expected value (mean): } & 2 \\\text { variance: } & - 1 \\\text { standard deviation: } & 4\end{array}$
E) $\begin{array} { l l } \text { expected value (mean): } & - 1 \\\text { variance: } & 2 \\\text { standard deviation: } & 4\end{array}$

Question 73

Evaluate using Pascal's triangle. Show your work.

Question 74

Find a formula for the sum of the n terms of the sequence. $\frac { 1 } { 5 } , \frac { 3 } { 25 } , \frac { 9 } { 125 } , \frac { 27 } { 625 } , \mathbf { K }$

A) $- \frac { 3 ^ { n } - 5 ^ { n } } { 2 \left( 5 ^ { n } \right) }$
B) $- \frac { 3 \left( 3 ^ { n } - 5 ^ { n } \right) } { 2 \left( 5 ^ { n } \right) }$
C) $\frac { 3 ^ { n - 1 } } { 5 ^ { n } }$
D) $\frac { 3 ^ { n } + 5 ^ { n } } { 8 ( 5 ) ^ { n } }$
E) $\frac { 1 } { 5 ^ { n } }$

Question 75

Prove the inequality for the indicated integer values of n.

Question 76

Use mathematical induction to prove the property for all positive integers n.

Question 77

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 9 } n ^ { 3 }$

A)1296
B)4050
C)729
D)285
E)2025

Question 78

Find the mean and median of $f ( x ) = \frac { 1 } { 17 } , [ 0,17 ]$ .

A) $\begin{array} { l l } \text { mean: } & 34.00 \\\text { median: } & 34.00\end{array}$
B) $\begin{array} { l l } \text { mean: } & 34.00 \\\text { median: } & 0.12\end{array}$
C) $\begin{array} { l l } \text { mean: } & 5.67 \\\text { median: } & 0.50\end{array}$
D) $\begin{array} { l l } \text { mean: } & 8.50 \\\text { median: } & 8.50\end{array}$
E) $\begin{array} { l l } \text { mean: } & 0.50 \\\text { median: } & 0.50\end{array}$

Question 79

Use mathematical induction to prove the formula for every positive integer n. Show all your work.

Question 80

Calculate the binomial coefficient: $7 C _ { 5 }$

A)2520
B)35
C)21
D)1
E)0