Question 1

Solve the problem.
-A tourist in France wants to visit 10 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order?

Accepted Answer

The total number of ways to arrange the visit to 10 cities is 10!, which is 3,628,800. There is only one way to visit the cities in alphabetical order, so the probability is $\frac{1}{3,628,800}$.

Question 2

If an apple is hanging from a string and three flies land on it, find the probability that all three are on points that are within the same hemisphere.

Accepted Answer

The probability that all three flies land within the same hemisphere of the apple is 0.25. This is because, regardless of where the first fly lands, the second and third flies must land within the same hemisphere as the first fly to satisfy the condition. The surface of the apple can be divided into two hemispheres with respect to the first fly's landing point. The probability that the second fly lands in the same hemisphere as the first is 1/2, and similarly, the probability for the third fly is also 1/2. Multiplying these probabilities gives 1/2 * 1/2 = 1/4 or 0.25.

Question 3

Find the indicated probability.
-A class has 11 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 answer of Find the indicated probability.
-A class has 11...

Question 4

11 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 formula for combinations: n(n-1)/2, where n is the number of wrestlers. Here, n=11, so the total number of matches is 11(11-1)/2 = 55.

Question 5

Solve the problem.
-In a certain lottery, five different numbers between 1 and 36 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 total number of ways to select 5 different numbers from 36 in a specific order is $36 \times 35 \times 34 \times 33 \times 32$, which equals 45,239,040. Therefore, the probability of selecting the correct 5 numbers in the correct order is $1$ out of $45,239,040$, or $\frac { 1 } { 45,239,040 }$.

Question 6

Solve the problem.
-There are 7 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 number of ways to choose and arrange 3 positions from 7 members is a permutation problem: 7P3 = 7 × 6 × 5 = 210.

Question 7

If you are told that a randomly selected mystery person was born in the 1990's, what is the probability of guessing his/her exact birth date (including year)?

Accepted Answer

The 1990s span from January 1, 1990, to December 31, 1999, totaling 10 years. Assuming each year has 365 days (ignoring leap years for simplicity), the total number of days in the 1990s is $10 \times 365 = 3650$ days. The probability of guessing the correct day is $1/3650$, which equals approximately $2.738 \times 10^{-4}$.

Question 8

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 n+0.39908993-0.43429448 \mathrm{n} $. Evaluate 79 ! using the approximation given above.

Accepted Answer

Using the given formula $ \mathrm{K}=(\mathrm{n}+0.5) \log n+0.39908993-0.43429448 \mathrm{n} $ for $ n = 79 $, we calculate $ K $ and then use $ 10^K $ to approximate $ 79! $. The calculation aligns closest with option A, $ 8.937 \times 10^{116} $, when following the approximation method provided.

Question 9

Solve the problem.
-A tourist in France wants to visit 7 different cities. How many different routes are possible?

Accepted Answer

The number of different routes possible is calculated by finding the factorial of the number of cities, which is 7 factorial (7!). 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040.

Question 10

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 5 questions on the survey?

Accepted Answer

The number of different surveys required to cover all possible arrangements of 5 questions is calculated by finding the factorial of 5, which is 5! = 5 × 4 × 3 × 2 × 1 = 120.

Quiz 4: Probability

Solve the problem. -A tourist in France wants to visit 10 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order?

If an apple is hanging from a string and three flies land on it, find the probability that all three are on points that are within the same hemisphere.

Find the indicated probability. -A class has 11 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.)

11 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. -In a certain lottery, five different numbers between 1 and 36 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. -There are 7 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?

If you are told that a randomly selected mystery person was born in the 1990's, what is the probability of guessing his/her exact birth date (including year) ?

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 $\mathrm{K}=(\mathrm{n}+0.5) \log n+0.39908993-0.43429448 \mathrm{n}$ . Evaluate 79 ! using the approximation given above.

Solve the problem. -A tourist in France wants to visit 7 different cities. How many different routes are possible?

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 5 questions on the survey?