Chapter 5: Sampling without Replacement

1
Chapter 5 Sampling without Replacement

• 5.1 Counting Formula
• Definition Factorial n is denoted by n! and
  given by
• for n positive integer. We define 0!1.
• Example 1

2
Section 5.1 Counting Formula

• Example 2 A box contains three chips labeled A
  and two chips labeled B. All five chips are
  selected without replacement from the box. Think
  of every possible ordered selection as a word.
  How many words are there?
• Solution

3
Section 5.1 Counting Formula

• Thus we see that each word has probability 1/10.
  Since exactly one of these words must appear,
• We must add 1/10 a total of ten times to get 1.
  Thus, the number of words with 3As, 2 Bs is

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Section 5.2Probabilities for Sampling without
Replacement

• Example 3 A box contains eight red and six white
  chips. Four chips are drawn at random without
  replacement. Let X denote the number of red chips
  drawn. Find an expression for the probabilities
  of the following events

5
Section 5.2Probabilities for Sampling without
Replacement

• Example 4 A box contains 10 chips numbered 1, 2,
  3,,10. Five chips are selected at random without
  replacement. Find an expression for the
  probability that the
• smallest number drawn is 4
• median number drawn is 4.
• Solution Let Eequal to 4, Ggreater than
  4, and Lless than 4.

6
Section 5.2Probabilities for Sampling without
Replacement

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Section 5.2Probabilities for Sampling without
Replacement

• Example 5 In a close election in a small town,
  637 people voted for candidate A compared to 630
  people for candidate B, a margin of seven votes.
  An investigation found that ten people who voted
  in the election should not have ( we dont know
  who they voted for). This is more than the margin
  of victory. What is the probability that the
  random removal of ten votes would reverse the
  election results ?

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Section 5.2Probabilities for Sampling without
Replacement

• Example 6 Each of two precincts has 100 voters.
  The number of Democrats in precinct I is 20 and
  in precinct II is 80. Assume the other voters are
  Republicans. For each of the following methods of
  selecting two voters, find the distribution of X,
  the number of Democrats selected.
• Select one voter at random from each precinct.
• Select two voters at random without replacement
  from the combined group of 200 voters.
• Select one of the two precincts at random, then
  select two voters at random without replacement
  from the selected precinct.

9
Section 5.2Probabilities for Sampling without
Replacement

• Solution (i)

10
Section 5.2Probabilities for Sampling without
Replacement

• Solution (ii)

11
Section 5.2Probabilities for Sampling without
Replacement

• Solution (iii)

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Section 5.2Probabilities for Sampling without
Replacement

• Example 7 We want to estimate the number of fish
  in a pond. We catch three fish at random (without
  replacement), tag them, and then release them
  back into the pond. A day later we catch five
  fish and we observe two of the five are tagged.
  Estimate the number of fish in the pond as the
  number which will maximize the probability of
  what actually observed.

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Section 5.2Probabilities for Sampling without
Replacement

• Solution Let T denote tagged, U untagged,
  and let X be the number of tagged fish drawn.
• Suppose there are five fish in the pond
• Since three fish were tagged, the
  contents of the pond are 3T, 2U. Next we catch
  five fish without replacement. P(X2)0 since it
  is not possible to get two tagged fish and three
  untagged fish from this pond.
• Suppose there are six fish in the pond
• Since three fish were tagged, the
  contents of the pond are 3T, 3U. Next we catch
  five fish without replacement.

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Section 5.2Probabilities for Sampling without
Replacement

• Suppose there are seven fish in the pond
• Since three fish were tagged, the
  contents of the pond are 3T, 4U. Next we catch
  five fish without replacement.

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Section 5.2Probabilities for Sampling without
Replacement

• Suppose there are eight fish in the pond
• Since three fish were tagged, the
  contents of the pond are 3T, 5U. Next we catch
  five fish without replacement.

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Section 5.2Probabilities for Sampling without
Replacement

• Suppose there are nine fish in the pond
• Since three fish were tagged, the
  contents of the pond are 3T, 6U. Next we catch
  five fish without replacement.

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Section 5.2Probabilities for Sampling without
Replacement

• Therefore, we estimate the number of fish in the
  pond to be seven.
• Page 127-130
• Do the following problems
• 1-2, 4, 8-10, 12, 14, 15, 17, 19