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Chapter 7: Problem 9

Explain the difference between sampling with replacement and sampling without replacement. Suppose you have a deck of 52 cards and want to select two cards. Describe both procedures.

Short Answer

Expert verified
Sampling with replacement refers to a method where a selected sample is returned back to the population for it to be possible choosing it again. Here, probabilities remain constant. On contrary, sampling without replacement means once a sample is selected, it's not returned back to the population affecting the probabilities of further draws.

Step by step solution

01

Explanation of Sampling with Replacement

In sampling with replacement, after drawing a card from a deck of 52 cards, it is returned back to the deck before drawing the second card. Consequently, the deck still remains with 52 cards for the second draw. Therefore, the probability of drawing any card remains the same for both draws, 1/52.
02

Explanation of Sampling Without Replacement

In sampling without replacement, once a card is drawn from the deck, it is not returned back. As a result, for the second draw, the deck only contains 51 cards. Subsequently, the probability of drawing any particular card on the second draw changes to 1/51.

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Key Concepts

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

Understanding Sampling with Replacement
Sampling with replacement is a method where every item in a population is put back into the population after being selected. Imagine you have a deck of 52 cards. When you draw a card, you note it down and then place it back into the deck before drawing the next card. This way, the deck remains unchanged for each draw.
This method ensures that each card has the same chance of being chosen every time you draw. The success probability for picking any one card randomly is always \(\frac{1}{52}\).
  • This is useful when you want each selection to be independent from the others.
  • It is often used in random sampling techniques in statistics.
  • Consequently, repeated outcomes are possible since the same item can be drawn multiple times.
Exploring Sampling Without Replacement
On the other hand, sampling without replacement involves selecting an item and not returning it to the population before the next draw. Let's return to the deck of 52 cards example. When you draw a card and do not put it back, you now have only 51 cards left in the deck for the next draw. Thus, the scenario is different for your second card.
With this method, you'll notice that the probability of drawing the second card changes to \(\frac{1}{51}\), as there is one less card.
  • This technique is useful when you want your selection to impact subsequent choices.
  • It's commonly used in situations where repeated outcomes are not possible or desired.
  • It reflects real-world situations more accurately in some cases, such as lottery draws or card games where once an item is selected, it can't be chosen again.
Grasping Probability
Probability is a measure of the likelihood of an event occurring. It ranges from 0 to 1, where 0 means the event is impossible, and 1 implies certainty. Thinking about our card deck example can help you grasp how probability works in practical scenarios.
In sampling with replacement, each draw has a \(\frac{1}{52}\) probability, highlighting that previous selections do not affect subsequent ones. This is a perfect illustration of independent events.
  • In contrast, sampling without replacement reveals a dynamic probability scenario.
  • As each card isn't returned, the sample space is reduced, altering probabilities with each draw.
  • Both scenarios showcase how different sampling methods can affect probability calculation and outcomes.
Understanding these differences not only helps in statistical calculations but also in practical applications where such probability assessments are crucial.

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