Mat 181 Chapter 4 Practice Problemsname ✓ Solved

MAT 181 Chapter 4 Practice Problems Name______________________________________ Date:________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the indicated probability. 1) Two 6-sided dice are rolled. What is the probability that the sum of the two numbers on the dice will be 4?

1) _____________ 2) A class consists of 66 women and 98 men. If a student is randomly selected, what is the probability that the student is a woman? 2) _____________ Answer the question, considering an event to be "unusual" if its probability is less than or equal to 0.05. 3) If you drew one card from a standard deck, would it be "unusual" to draw an eight of clubs? 3) _____________ From the information provided, create the sample space of possible outcomes.

4) Flip a coin three times. 4) _____________ Answer the question. 5) Find the odds against correctly guessing the answer to a multiple choice question with 6 possible answers. 5) _____________ 6) In a certain town, 25% of people commute to work by bicycle. If a person is selected randomly from the town, what are the odds against selecting someone who commutes by bicycle?

6) _____________ Find the indicated probability. 7) A spinner has equal regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3? 7) _____________ 8) A study conducted at a certain college shows that 51% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that 5 randomly selected graduates all find jobs in their chosen field within a year of graduating.

Round to the nearest thousandth if necessary. 8) _____________ 9) A IRS auditor randomly selects 3 tax returns from 45 returns of which 15 contain errors. What is the probability that she selects none of those containing errors? Round to four decimal places. 9) _____________ 10) The table below describes the smoking habits of a group of asthma sufferers.

If one of the 1127 people is randomly selected, find the probability that the person is a man or a heavy smoker. 10) _____________ 11) A bag contains 6 red marbles, 3 blue marbles, and 1 green marble. Find P (not blue). 11) _____________ 12) A batch consists of 12 defective coils and 88 good ones. Find the probability of getting two good coils when two coils are randomly selected if the first selection is replaced before the second is made.

12) _____________ Find the indicated probability. Round to the nearest thousandth. 13) An unprepared student makes random guesses for the ten true-false questions on a quiz. Find the probability that there is at least one correct answer. 13) _____________ Find the indicated probability.

14) A IRS auditor randomly selects 3 tax returns from 49 returns of which 7 contain errors. What is the probability that she selects none of those containing errors? Round to four decimal places. 14) _____________ Find the indicated probability. Express your answer as a simplified fraction unless otherwise noted.

15) The following table contains data from a study of two airlines which fly to Small Town, USA. If one of the 87 flights is randomly selected, find the probability that the flight selected is an Upstate Airlines flight given that it was late. 15) _____________ Find the indicated probability. Round to the nearest thousandth. 16) A sample of 4 different calculators is randomly selected from a group containing 18 that are defective and 40 that have no defects.

What is the probability that at least one of the calculators is defective? 16) _____________ Solve the problem. 17) How many ways can an IRS auditor select 5 of 10 tax returns for an audit? 17) _____________ Solve the problem. 18) A state lottery involves the random selection of six different numbers between 1 and 31.

If you select one six number combination, what is the probability that it will be the winning combination? 18) _____________ Solve the problem. 19) 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? 19) _____________ 1

Paper for above instructions

MAT 181 Chapter 4 Practice Problems Solutions

This document provides solutions to the problems from Chapter 4 of MAT 181, concentrating on probability, combinations, and fundamental principles of counting.

Short Answers

1. Two 6-sided dice are rolled. What is the probability that the sum of the two numbers on the dice will be 4?
- The possible outcomes for two dice rolling a sum of 4 are: (1, 3), (2, 2), and (3, 1). There are a total of 36 outcomes (6 sides × 6 sides). Therefore, the probability is:
\[
P(\text{Sum of 4}) = \frac{3}{36} = \frac{1}{12} \approx 0.0833
\]
2. A class consists of 66 women and 98 men. If a student is randomly selected, what is the probability that the student is a woman?
- Total students = 66 + 98 = 164
\[
P(\text{Woman}) = \frac{66}{164} = \frac{33}{82} \approx 0.4024
\]
3. If you drew one card from a standard deck, would it be "unusual" to draw an eight of clubs?
- There are 52 cards in a deck, and drawing one specific card (eight of clubs) has a probability of:
\[
P(\text{Eight of Clubs}) = \frac{1}{52} \approx 0.0192
\]
Since 0.0192 < 0.05, it is unusual.
4. Flip a coin three times.
- The sample space (S) can be represented as:
\[
S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}
\]
5. Find the odds against correctly guessing the answer to a multiple choice question with 6 possible answers.
- If the chance of guessing correctly is \( \frac{1}{6} \), the odds against are:
\[
\text{Odds against} = \frac{5}{1}
\]
6. In a certain town, 25% of people commute to work by bicycle. What are the odds against selecting someone who commutes by bicycle?
- The probability of not commuting by bicycle is 75% or \( \frac{3}{4} \). Thus, the odds against are:
\[
\text{Odds against} = \frac{3}{1}
\]
7. A spinner has equal regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3?
- Even numbers = 10 (2, 4, ..., 20)
- Multiples of 3 = 7 (3, 6, 9, 12, 15, 18, 21)
- Even multiples of 3 = 3 (6, 12, 18)
- Total even or multiple of 3 = 10 + 7 - 3 = 14
\[
P(\text{Even or Multiple of 3}) = \frac{14}{21} = \frac{2}{3} \approx 0.6667
\]
8. 51% of the graduates find a job in their chosen field within a year after graduation. Find the probability that 5 randomly selected graduates all find jobs.
\[
P(\text{All find jobs}) = (0.51)^5 \approx 0.0379
\]
9. An IRS auditor randomly selects 3 tax returns from 45 returns of which 15 contain errors. What is the probability that she selects none of those containing errors?
- Total combinations of selecting 3 from 45:
\[
C(45, 3) = \frac{45!}{3!(45-3)!} = 14190
\]
- Combinations of selecting 3 from 30 (returns without errors):
\[
C(30, 3) = \frac{30!}{3!(30-3)!} = 4060
\]
\[
P(\text{No Errors}) = \frac{C(30, 3)}{C(45, 3)} = \frac{4060}{14190} \approx 0.2861
\]
10. Find the probability that a person is a man or a heavy smoker, given the provided data. Assume the total number of individuals is 1127.
- Let’s denote:
- Men = \( M \)
- Heavy Smokers = \( S \)
- Use data to derive \( P(M \cup S) = P(M) + P(S) - P(M \cap S) \)
- Without specific numbers for men and heavy smokers, we cannot calculate.
11. A bag contains 6 red marbles, 3 blue marbles, and 1 green marble. Find P(not blue).
- Total marbles = 6 + 3 + 1 = 10. Non-blue marbles = 7 (6 red + 1 green).
\[
P(\text{Not Blue}) = \frac{7}{10} = 0.7
\]
12. A batch consists of 12 defective coils and 88 good ones. Find the probability of getting two good coils when two coils are randomly selected if the first selection is replaced before the second is made.
\[
P(2 \text{ Good}) = \left(\frac{88}{100}\right) \times \left(\frac{88}{100}\right) = 0.7744
\]
13. An unprepared student makes random guesses for the ten true-false questions. Find the probability that there is at least one correct answer.
- Total possibilities = \( 2^{10} = 1024 \)
- Probability of getting none correct = \( \left(\frac{1}{2}\right)^{10} = \frac{1}{1024} \)
\[
P(\text{At least 1 correct}) = 1 - \frac{1}{1024} = 0.9990
\]
14. An IRS auditor randomly selects 3 tax returns from 49 returns of which 7 contain errors. Find the probability that she selects none of those containing errors.
- Total combinations of selecting 3 from 49:
\[
C(49, 3) = \frac{49!}{3!(46)!} = 18424
\]
- Combinations of selecting 3 from 42 (returns without errors):
\[
C(42, 3) = \frac{42!}{3!(39)!} = 11480
\]
\[
P(\text{No Errors in 3}) = \frac{C(42, 3)}{C(49, 3)} \approx 0.6223
\]
15. Probability that the flight selected is an Upstate Airlines flight given it was late will depend on the number of delayed flights. Without specific data, we cannot calculate.
16. What is the probability that at least one of the selected calculators is defective given 18 defective and 40 good?
- Using complementary probability,
\[
P(\text{No Defect}) = \frac{40}{58} \times \frac{39}{57} \times \frac{38}{56} \times \frac{37}{55} \quad P(\text{At least 1 defect}) = 1 - P(\text{No Defect}) \approx 0.9326
\]
17. How many ways can an IRS auditor select 5 of 10 tax returns for an audit?
\[
C(10, 5) = \frac{10!}{5!5!} = 252
\]
18. If you select one six-number combination in a lottery with numbers 1 to 31, the probability of choosing the winning combination is:
\[
C(31, 6) = \frac{31!}{6!(31-6)!} = 593775
\]
\[
P(\text{Winning}) = \frac{1}{593775} \approx 0.000001684
\]
19. There are 6 members on a board of directors. The number of ways to elect a chairperson, a secretary, and a treasurer is calculated by:
- There are nesting choices: 6 options for Chair, 5 for Secretary, and 4 for Treasurer. Thus:
\[
6 \times 5 \times 4 = 120
\]

References

1. Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
2. Moore, D. S., & McCabe, G. P. (2006). Introduction to the Practice of Statistics. W.H. Freeman.
3. Bluman, A. G. (2018). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
4. Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
5. Devore, J. L. (2015). Probability and Statistics. Cengage Learning.
6. Walpole, R. E., Myers, R. M., Myers, S. L., & Ye, K. (2012). Probability & Statistics. Pearson.
7. Sullivan, M. (2019). Statistics. Pearson.
8. Siegel, A. F., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
9. Triola, M. F. (2018). Elementary Statistics. Pearson.
10. Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A Nonparametric Approach to Statistical Inference. Sage Publications.