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Chapter 5: Problem 35

A university found that \(20 \%\) of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. a. Compute the probability that two or fewer will withdraw. b. Compute the probability that exactly four will withdraw. c. Compute the probability that more than three will withdraw. d. Compute the expected number of withdrawals.

Short Answer

Expert verified
a. 0.2061; b. 0.2182; c. 0.5885; d. 4

Step by step solution

01

Define the random variable

Let \( X \) represent the number of students who withdraw from the course. Since each student has a \(20\%\) chance of withdrawing, we can model this situation as a binomial distribution where the number of trials \( n = 20 \), and the probability of success \( p = 0.2 \).
02

Formula for Binomial Distribution Probability

The probability of \( X = x \) successes in a binomial distribution is given by the formula: \[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\]where \( \binom{n}{x} \) is the binomial coefficient.
03

Compute Probability for Part a

Part a requires calculating the probability that two or fewer students will withdraw: \[P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)\]Compute each probability using the binomial formula:\[P(X = 0) = \binom{20}{0} (0.2)^0 (0.8)^{20} = 0.0115\]\[P(X = 1) = \binom{20}{1} (0.2)^1 (0.8)^{19} = 0.0577\]\[P(X = 2) = \binom{20}{2} (0.2)^2 (0.8)^{18} = 0.1369\]Thus,\[P(X \leq 2) = 0.0115 + 0.0577 + 0.1369 = 0.2061\]
04

Compute Probability for Part b

For exactly four students withdrawing, use the binomial formula:\[P(X = 4) = \binom{20}{4} (0.2)^4 (0.8)^{16} = 0.2182\]
05

Compute Probability for Part c

To find the probability that more than three students withdraw, compute:\[P(X > 3) = 1 - P(X \leq 3)\]We already have \( P(X \leq 2) \), so calculate:\[P(X = 3) = \binom{20}{3} (0.2)^3 (0.8)^{17} = 0.2054\]Then,\[P(X \leq 3) = P(X \leq 2) + P(X = 3) = 0.2061 + 0.2054 = 0.4115\]Thus,\[P(X > 3) = 1 - 0.4115 = 0.5885\]
06

Compute Expected Number of Withdrawals

The expected number of withdrawals in a binomial distribution is given by:\[E(X) = n \cdot p = 20 \cdot 0.2 = 4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is a branch of mathematics concerned with the analysis of random phenomena. Some events in life don't occur with certainty. Instead, they occur with certain probabilities. Probability is the measure of the likelihood that an event will occur.

For the situation with students withdrawing from a statistics course, this scenario is best modeled using the binomial distribution. This particular distribution is used when there are fixed numbers of trials, and each trial has only two possible outcomes: 'success' or 'failure'. For this exercise, suppose a 'success' means a student withdraws. Each student, therefore, has a fixed probability of 20% of withdrawing.
  • The number of trials is fixed, in this case, 20 students.
  • Each trial (each student) is independent of the others.
  • The probability of success (a student withdrawing) remains constant, defined here as 0.2 or 20%.
In our case, we look at different probabilities for numbers of students withdrawing, like none, two, four, etc., which are solved by applying the binomial probability formula.
Statistics Education
Statistics education helps students understand data analysis, probability, and the principles guiding decisions made under uncertainty. This binomial distribution exercise is a part of statistics education, providing practical understanding through real-world scenarios.

When students deal with this problem, they gain exposure to essential concepts in statistics such as:
  • Probability Calculations: Learning to compute probabilities, for example, the likelihood of zero, two, or four students withdrawing from the course.
  • Interpreting Results: Making sense of calculated probabilities in the context of real-world situations. For instance, understanding that a 20% chance per student over many trials implies some will withdraw.
  • Critical Thinking: Deciding what meaningful conclusions can be drawn from the analysis, such as how many students the course might expect to lose week to week.
Statistics education equips students with the skill to turn data into insightful interpretations, using established mathematical frameworks like the one conveyed by the binomial distribution.
Expected Value
The expected value is a concept from probability theory that gives us a measure of the average or central tendency of random variables. It's like predicting the long-term average outcome if we repeat the process many times.

In the context of our university students' scenario, the expected number of students withdrawing from the course can be calculated using the binomial distribution's expected value formula:
  • Formula: The expected value is computed as \(E(X) = n \cdot p\).
  • Application: In this exercise, with \(n = 20\) (the 20 students) and \(p = 0.2\) (20% withdrawal rate), the expected number of students withdrawing is 4.
  • Interpretation: This means that, on average, we expect 4 out of the 20 students to withdraw from the course each time this situation occurs.
Understanding the expected value helps in planning and resource allocation, providing a predictive figure for likely outcomes.

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Most popular questions from this chapter

A survey showed that the average commuter spends about 26 minutes on a one-way doorto-door trip from home to work. In addition, \(5 \%\) of commuters reported a one-way commute of more than one hour (Bureau of Transportation Statistics website, January 12,2004 ). a. If 20 commuters are surveyed on a particular day, what is the probability that 3 will report a one-way commute of more than one hour? b. If 20 commuters are surveyed on a particular day, what is the probability that none will report a one-way commute of more than one hour? c. If a company has 2000 employees, what is the expected number of employees who have a one-way commute of more than one hour? d. If a company has 2000 employees, what are the variance and standard deviation of the number of employees who have a one-way commute of more than one hour?

Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a warning. Assume that a particular detection system has a .90 probability of detecting a missile attack. Use the binomial probability distribution to answer the following questions. a. What is the probability that a single detection system will detect an attack? b. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack? c. If three systems are installed, what is the probability that at least one of the systems will detect the attack? d. Would you recommend that multiple detection systems be used? Explain.

The National Safety Council (NSC) estimates that off-the-job accidents cost U.S. businesses almost \(\$ 200\) billion annually in lost productivity (National Safety Council, March 2006 ). Based on NSC estimates, companies with 50 employees are expected to average three employee off-the-job accidents per year. Answer the following questions for companies with 50 employees. a. What is the probability of no off-the-job accidents during a one-year period? b. What is the probability of at least two off-the-job accidents during a one- year period? c. What is the expected number of off-the-job accidents during six months? d. What is the probability of no off-the-job accidents during the next six months?

The Zagat Restaurant Survey provides food, decor, and service ratings for some of the top restaurants across the United States. For 15 restaurants located in Boston, the average price of a dinner, including one drink and tip, was \(\$ 48.60 .\) You are leaving for a business trip to Boston and will eat dinner at three of these restaurants. Your company will reimburse you for a maximum of \(\$ 50\) per dinner. Business associates familiar with these restaurants have told you that the meal cost at one-third of these restaurants will exceed \(\$ 50 .\) Suppose that you randomly select three of these restaurants for dinner. a. What is the probability that none of the meals will exceed the cost covered by your company? b. What is the probability that one of the meals will exceed the cost covered by your company? c. What is the probability that two of the meals will exceed the cost covered by your company? d. What is the probability that all three of the meals will exceed the cost covered by your company?

An average of 15 aircraft accidents occur each year (The World Almanac and Book of Facts, 2004 ). a. Compute the mean number of aircraft accidents per month. b. Compute the probability of no accidents during a month. c. Compute the probability of exactly one accident during a month. d. Compute the probability of more than one accident during a month.
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