Probability problems

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Problems need to include all required steps and answer(s) for full credit. All answers need to be reduced to lowest terms where possible. If the answer is in %, show two decimal places.
Answer the following problems showing your work and explaining (or analyzing) your results. Submit your work in a typed Microsoft Word document.
1. During an economic downturn, Lanier Company is forced to lay off employees. The table below breaks down the layoffs:
2. Being Laid Off Not Being Laid Off Total
Managers 50 235 285
Non-Managers 125 575 700
Total 175 810 985
a. What is the probability that an employee of Lanier is being laid off given that she/he is a non-manager? (3 pts)
b. What is the probability that an employee selected at random is a manager? (3 pts)
3. The following data set represents the average day temperature in Orange County, Calif., for 20 days in December.

Degrees (in Fahrenheit)
43 65 55 49 60 52 45 61 59 47
62 46 65 57 44 64 53 48 63 56
4. Find the probability that a randomly selected day will have an average temperature of at least 60 ℉. (4 pts)
5. A survey was taken to determine the birthplace of a class of students in a statistics class. Below is the data set.
Gender Number of Students Birthplace
Male
Female 6
10 San Diego
San Diego
Male
Female 12
8 Los Angeles
Los Angeles
Male
Female 5
9 San Francisco
San Francisco
a. What is the probability that a student was born in Los Angeles given the student was male? (3 pts)
b. What is the probability a student was born in San Diego or the student was female? (3 pts)
6. What is the probability of throwing one die and getting a number greater than 5? (4pts)

7. Two coins are tossed 100 times. Below is the data set:
Two heads 29
One heads 45
No heads 26
a. What is the probability of getting two heads? (3 pts)
b. What is the probability of getting no heads? (3 pts)
8. Broad Tire Manufacturing Company kept a record of the distance before a tire needed to be replaced. The table shows the results of 500 tires.
Distance in miles Less than 2,500 2,500–5,500 5,501–8,700 More than 8,700
Frequency 10 105 160 225
a. If a tire is bought from this company, what is the probability that it must be replaced with less than 2,500 miles? (3 pts)
b. What is the probability that it will last more than 5,500 miles? (3 pts)
9. Two golfers, Phillips and Woodson, play a golf match. The probability of Phillips winning the match is 56%. What is the probability of Woodson winning the match? (4pts)

10. The percentage of grades obtained by a student in five tests is given below:
Test 1 2 3 4 5
% of Grades 65 80 73 79 59
11. Find the probability of the student getting at least 70% on a test. (4 pts)
12. Find the probability of correctly answering the first 5 questions on a multiple- choice test by randomly guessing the answer. Each question has 4 possible answers. (4 pts)
13. One card is drawn from a deck of 52 cards, well-shuffled.
a. Calculate the probability that the card will be an ace or a king. (3 pts)
b. Calculate the probability the card will be a queen. (3pts)

 

Sample Solution

Problem 1:

a. Probability of being laid off given non-manager: (Number of non-managers laid off) / (Total number of non-managers) = 125 / 700 = 5/28

b. Probability of being a manager: (Number of managers) / (Total number of employees) = 285 / 985 = 95/325

Problem 2:

Explanation: We need to calculate the frequency of days with temperatures at least 60 ℉ and divide it by the total number of days.

Calculations:

  • Count the number of days with temperatures >= 60: 9 days
  • Calculate the probability: 9 days / 20 days = 0.45 or 45%

Problem 3:

a. Probability of Los Angeles given male: (Number of male students born in LA) / (Total number of male students) = 12 / 21 = 4/7

b. Probability of San Diego or female: (Number of students born in SD) + (Number of female students) – (Number of female students born in SD) / Total number of students = 6 + 8 – 5 / 25 = 9 / 25

Problem 4:

Explanation: There are 6 outcomes (1, 2, 3, 4, 5, 6), and 1 outcome is favorable (greater than 5).

Calculation: Probability = Favorable outcomes / Total outcomes = 1 / 6 = 0.167 or 16.7%

Problem 5:

a. Probability of two heads: Favorable outcomes / Total outcomes = 29 / 100 = 29/100

b. Probability of no heads: Favorable outcomes / Total outcomes = 26 / 100 = 13/50

Problem 6:

a. Probability of less than 2,500 miles: Frequency of < 2,500 / Total frequency = 10 / 500 = 1/50

b. Probability of more than 5,500 miles: Frequency of > 5,500 / Total frequency = 225 + 160 / 500 = 19/25

Problem 7:

Explanation: The probability of Woodson winning is the complement of the probability of Phillips winning.

Calculation: Probability of Woodson winning = 1 – Probability of Phillips winning = 1 – 0.56 = 0.44 or 44%

Problem 8:

Explanation: We need to count the number of grades 70% or higher and divide it by the total number of grades.

Calculations:

  • Count the number of grades >= 70: 3 grades (tests 2, 4, and 5)
  • Calculate the probability: 3 grades / 5 tests = 0.6 or 60%

Problem 9:

Explanation: There are 4 ways to get the first 5 answers correct by guessing (1/4 for each question).

Calculation: Probability = (1/4)^5 = 1/1024

Problem 10:

Explanation: There are 4 aces and 4 kings in a deck, and 13 cards of each suit.

a. Probability of ace or king: (Number of aces + number of kings) / Total number of cards = (4 + 4) / 52 = 8/52 = 2/13

b. Probability of queen: Number of queens / Total number of cards = 4 / 52 = 1/13

I hope these solutions and explanations are helpful!

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