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Chapter 5: Problem 69

Suppose that you have just received a shipment of 20 modems. Although you don鈥檛 know this, 3 of the modems are defective. To determine whether you will accept the shipment, you randomly select 4 modems and test them. If all 4 modems work, you accept the shipment. Otherwise, the shipment is rejected. What is the probability of accepting the shipment?

Short Answer

Expert verified
The probability of accepting the shipment is 0.491.

Step by step solution

01

- Understand the Problem

We need to find the probability of accepting a shipment of 20 modems when selecting 4 modems at random and testing them. The shipment is accepted only if all 4 modems tested are non-defective.
02

- Total Number of Ways to Choose 4 Modems

First, calculate the total number of ways to choose 4 modems out of 20. This can be done using the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Where: - \(n\) is the total number of modems (20) - \(k\) is the number of modems to be chosen (4) \[ \binom{20}{4} = \frac{20!}{4!(20-4)!} = 4845 \]
03

- Number of Ways to Choose 4 Non-Defective Modems

Next, calculate the number of ways to choose 4 non-defective modems from the 17 non-defective modems. This is again done using the combination formula: \[ \binom{17}{4} = \frac{17!}{4!(17-4)!} = 2380 \]
04

- Calculate the Probability

To find the probability that all 4 modems chosen are non-defective, divide the number of ways to choose 4 non-defective modems by the total number of ways to choose 4 modems: \[ P(\text{All 4 modems work}) = \frac{\binom{17}{4}}{\binom{20}{4}} = \frac{2380}{4845} \ \therefore P(\text{Accept the shipment}) = 0.491 \]

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Key Concepts

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

combinatorial analysis
Combinatorial analysis involves the study of counting, arrangement, and combination of objects. It's a fundamental part of probability theory. Here, we analyze different ways to select and arrange objects according to specific rules.
In our example of modems, we use combinatorial analysis to count how many ways we can choose 4 modems out of 20. This helps us determine the total number of possible outcomes.
Overall, combinatorial analysis is crucial for solving problems involving selection and arrangement, making it easier to calculate probabilities accurately.
binomial coefficient
The binomial coefficient, denoted as \(\binom{n}{k}\), is a key element in calculating combinations. It tells us how many ways we can choose \(k\) items from \(n\) items without considering the order.
In our example with modems, we use the binomial coefficient to find out how many ways we can choose 4 modems from 20. The formula for binomial coefficient is: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
  • \(n!\) represents the factorial of n
  • \(k\) is the number of items chosen
  • The denominator \((n-k)!\) accounts for the items not chosen
Using the binomial coefficient helps simplify complex counting problems, enabling easier probability calculations.
sampling without replacement
Sampling without replacement means selecting items from a population where each item can only be chosen once. Once selected, the item is not returned back into the population.
In our modem example, we randomly select 4 modems from a shipment of 20 without putting them back. This type of sampling affects the probabilities because the number of available items decreases with each selection.
It's important to differentiate between sampling with and without replacement when calculating probabilities, as it greatly impacts the outcome. In our problem, since defective modems are not replaced, it accurately reflects real-world scenarios where items are not reused.

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Most popular questions from this chapter

According to a Gallup Poll, about \(17 \%\) of adult Americans bet on professional sports. Census data indicate that \(48.4 \%\) of the adult population in the United States is male. (a) Assuming that betting is independent of gender, compute the probability that an American adult selected at random is male and bets on professional sports. (b) Using the result in part (a), compute the probability that an American adult selected at random is male or bets on professional sports. (c) The Gallup poll data indicated that \(10.6 \%\) of adults in the United States are males and bet on professional sports. What does this indicate about the assumption in part (a)? (d) How will the information in part (c) affect the probability you computed in part (b)?

Explain the Law of Large Numbers. How does this law apply to gambling casinos?

About \(13 \%\) of the population is left-handed. If two people are randomly selected, what is the probability that both are left-handed? What is the probability that at least one is right-handed?

Suppose that Ralph gets a strike when bowling \(30 \%\) of the time. (a) What is the probability that Ralph gets two strikes in a row? (b) What is the probability that Ralph gets a turkey (three strikes in a row)? (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a turkey, but fails to get a clover (four strikes in a row).

In seven-card stud poker, a player is dealt seven cards. The probability that the player is dealt two cards of the same value and five other cards of different value so that the player has a pair is \(0.44 .\) Explain what this probability means. If you play seven-card stud 100 times, will you be dealt a pair exactly 44 times? Why or why not?
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