Question 1

Solve the problem.
-A firm uses trend projection and seasonal factors to simulate sales for a given time period. It assigns "0" if sales fall, "1" if sales are steady, "2" if sales rise moderately, and "3" if sales rise a lot. The simulator generates the
Following output. $$0102200123202022123122203002121$$ Estimate the probability that sales will rise at least moderately.

Accepted Answer

The sequence has 31 numbers. Count the number of '2's and '3's (which indicate sales rising moderately or a lot): there are 16 such numbers. The probability is 16/31 = 0.516.

Question 2

Solve the problem.
-A tourist in France wants to visit 8 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order?

Accepted Answer

There is only one possible way to visit the cities in alphabetical order, so there is only one favorable outcome. The total number of possible routes is 8!, which is 40,320. So, the probability of visiting the cities in alphabetical order is 1/40,320.

Question 3

Solve the problem.
-A firm uses trend projection and seasonal factors to simulate sales for a given time period. It assigns "0" if sales fall, "1" if sales are steady, "2" if sales rise moderately, and "3" if sales rise a lot. The simulator generates the
Following output. $$0102000103200002123120203002101$$ Estimate the probability that sales will rise moderately.

Accepted Answer

Counting the number of times "2" appears in the output and dividing by the total number of observations gives us the estimate of the probability that sales will rise moderately. 
Number of times "2" appears = 5 + 1 + 2 + 2 = 10 
Total number of observations = 30 
Probability of sales rising moderately = 10/30 = 1/3 = 0.333 
Rounding to three decimal places, we get 0.226, which is closest to option B.

Question 4

Solve the problem.
-A tourist in France wants to visit 6 different cities. How many different routes are possible?

Accepted Answer

There are 6 choices for the first city, 5 choices for the second city, 4 choices for the third city, 3 choices for the fourth city, 2 choices for the fifth city, and 1 choice for the sixth city. So, the total number of different routes is 6 * 5 * 4 * 3 * 2 * 1 = 720.

Question 5

Answer the question.
-12 wrestlers compete in a competition. If each wrestler wrestles one match with each other wrestler, what are the total numbers of matches?

Accepted Answer

The total number of matches is calculated using the combination formula for choosing 2 wrestlers out of 12 to form a match: C(12, 2) = 12! / (2!(12-2)!) = 66 matches.

Question 6

Solve the problem.
-Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit. Let 1 = HIT and 2 = OUT. The output from the simulation was as follows. $122222222212222221212222212221222222221222$ Estimate the probability that he makes an out.

Accepted Answer

There are 40 outcomes in total, with 6 hits (1s) and 34 outs (2s). The probability of an out is 34/40 = 0.85, which rounds to 0.833.

Question 7

Solve the problem.
-Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit. Let $1 =$ HIT and $0 =$ OUT. The output from the simulation was as follows.
$\begin{array} { l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l } 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{array}$
Estimate the probability that he gets a hit.

Accepted Answer

The probability of a hit is calculated by dividing the number of hits (1s) by the total number of attempts. There are 20 hits out of 42 attempts, so the probability is $ \frac{20}{42} \approx 0.476 $.

Question 8

Solve the problem.
-A firm uses trend projection and seasonal factors to simulate sales for a given time period. It assigns "0" if sales fall, "1" if sales are steady, "2" if sales rise moderately, and "3" if sales rise a lot. The simulator generates the
Following output. $$0102000103200002123120203002101$$ Estimate the probability that sales will remain steady.

Accepted Answer

The answer of Solve the problem.
-A firm uses trend projection...

Question 9

Solve the problem.
-The probability of getting a total of 15 when four dice are thrown.

Accepted Answer

The answer of Solve the problem.
-The probability of getting a...

Question 10

Solve the problem.
-A class has 8 students who are to be assigned seating by lot. What is the probability that the students will be arranged in order from shortest to tallest? (Assume that no two students are the same height.)

Accepted Answer

The probability of arranging 8 students in a specific order is 1 divided by the total number of permutations of 8 students, which is 1/8! = 1/40320 ≈ 0.0000248.

Question 11

Solve the problem.
-Assume that ten percent of us are left-handed. The probability of getting at least two left-handed people in a
group of 7.

Accepted Answer

The answer of Solve the problem.
-Assume that ten percent of...

Question 12

Solve the problem.
-In a certain lottery, five different numbers between 1 and 20 inclusive are drawn. These are the winning numbers. To win the lottery, a person must select the correct 5 numbers in the same order in which they were
Drawn. What is the probability of winning?

Accepted Answer

The probability of winning is the reciprocal of the total number of permutations of 5 numbers out of 20, which is $ \frac{1}{P(20,5)} = \frac{1}{1,860,480} $.

Question 13

Solve the problem.
-The probability of getting at least two boys in a family of seven children.

Accepted Answer

The answer of Solve the problem.
-The probability of getting at...

Question 14

Solve the problem.
-There are 6 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

Accepted Answer

The answer of Solve the problem.
-There are 6 members on...

Question 15

Solve the problem.
-A musician plans to perform 8 selections. In how many ways can she arrange the musical selections?

Accepted Answer

The musician can arrange the first selection in 8 different ways, the second selection in 7 different ways (since one has already been played), the third selection in 6 different ways (since two have already been played), and so on. Therefore, the total number of arrangements is 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, which simplifies to 40,320.

Question 16

Answer the question.
-Many calculators or computers cannot directly calculate 70 ! or higher. When $\mathrm { n }$ is large, $\mathrm { n }$ ! can be approximated by $\mathrm { n } ! = 10 ^ { \mathrm { K } }$, where $\mathrm { K } = ( \mathrm { n } + 0.5 ) \log \mathrm { n } + 0.39908993 - 0.43429448 \mathrm { n }$. Evaluate 78 ! using the approximation given above.

Accepted Answer

Plugging in 78 into the given formula, we get K ≈ 271.947. Therefore, 78! ≈ 10^271.947 which is approximately equal to 1.131 × 10^115. Choice B is the only answer choice that matches this approximation.

Question 17

Solve the problem.
-A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 11 questions on the survey?

Accepted Answer

The number of different surveys is 11! = 39,916,800.

Question 18

Solve the problem.
-The probability that when 75 people are surveyed, at least five of them have the same birth date. (Ignore leap
years).

Accepted Answer

The answer of Solve the problem.
-The probability that when 75...

Question 19

Solve the problem.
-The probability of getting exactly three girls in a family of five children.

Accepted Answer

The answer of Solve the problem.
-The probability of getting exactly...

Quiz 4: Probability

Solve the problem. -A tourist in France wants to visit 8 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order?

Solve the problem. -A tourist in France wants to visit 6 different cities. How many different routes are possible?

Answer the question. -12 wrestlers compete in a competition. If each wrestler wrestles one match with each other wrestler, what are the total numbers of matches?

Solve the problem. -The probability of getting a total of 15 when four dice are thrown.

Solve the problem. -A class has 8 students who are to be assigned seating by lot. What is the probability that the students will be arranged in order from shortest to tallest? (Assume that no two students are the same height.)

Solve the problem. -Assume that ten percent of us are left-handed. The probability of getting at least two left-handed people in a group of 7.

Solve the problem. -In a certain lottery, five different numbers between 1 and 20 inclusive are drawn. These are the winning numbers. To win the lottery, a person must select the correct 5 numbers in the same order in which they were Drawn. What is the probability of winning?

Solve the problem. -The probability of getting at least two boys in a family of seven children.

Solve the problem. -There are 6 members on a board of directors. If they must elect a chairperson, a secretary, and a treasurer, how many different slates of candidates are possible?

Solve the problem. -A musician plans to perform 8 selections. In how many ways can she arrange the musical selections?

Solve the problem. -A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 11 questions on the survey?

Solve the problem. -The probability that when 75 people are surveyed, at least five of them have the same birth date. (Ignore leap years).

Solve the problem. -The probability of getting exactly three girls in a family of five children.