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Learning intentions:
In this section we will examine:
- The process of sampling without replacement from small populations.
- The process of sampling without replacement from large populations.
Sampling without replacement
Sampling without replacement involves selecting an object from a finite group of objects and removing it from future selections. As we have removed an object future selections are made from a reduced sample space meaning the probabilities change between selections.
Sampling from small populations
When we sample from small populations, we can use a tree diagram to represent the sample space and determine the probabilities of events from the tree diagram.
15B - VIDEO EXAMPLE 1:
Suppose we have a bag containing 4 red and 6 green marbles. Two marbles are drawn randomly and they are not replaced before the next selection is made. Let X be the number of red marbles selected.
- Determine the probability distribution of the discrete random variable X.
- Find the probability that one or more red marbles is selected.
This skill was tested in VCAA 2015 Exam 2 - Section 1, Question 12.
Sampling from large populations
When we sample from large populations, we can use our knowledge of combinations to help determine probabilities without necessarily developing the entire probability distribution.
15B - VIDEO EXAMPLE 2:
A school is able to send 5 representatives to an international conference. 5 boys and 8 girls nominate to attend. Find the probability that at least 2 boys are selected to go to the conference.
Success criteria:
You will be successful if you can:
- Solve sampling problems without replacement involving small populations where a tree diagram can be used.
- Solve sampling problems without replacement involving large populations where combinations are required.
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