Section 6.1: Sampling with Replacement

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Section 6.1 Sampling with Replacement

• Binomial model.
• Example 1 A box contains eight red and six white
  chips. Four chips are drawn at random with
  replacement. Let X denote the number of red chips
  drawn. Find an expression for the probabilities
  of the following events

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Section 6.1 Sampling with Replacement

• Solution

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Section 6.1Sampling with Replacement

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Section 6.1Sampling with Replacement

• Binomial model.
• In terms of an experiment where chips
  are drawn from a box, the
  binomial model satisfies three conditions
• Two types of chips in the box.
• Fixed number of draws from the same box.
• Independent from draw to draw.

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Section 6.1 Sampling with Replacement

• Example 2Suppose 60 is a passing grade on a
  test. If there are five questions on the test and
  a student guesses on every question on the test,
  what is the probability that she passes?
• Assume a true-false test.
• Assume a multiple choice test with four
  alternatives for each question.

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Section 6.1 Sampling with Replacement

• Solution Since there are five questions on the
  test, a passing grade of 60 corresponds to
  getting at least three questions correct. Let C
  denote a correct answer and I an incorrect
  answer. For a true-false test, P(C).5 and
  P(I).5.
• Note For the binomial model, we think of drawing
  of five chips with
• replacement from a box containing one C
  and one I. Each draw corresponds to one question
  and the probability of getting a C (correct
  answer) is always 0.5.

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Section 6.1 Sampling with Replacement

• (ii) Since she is guessing and there are four
  alternatives for each question, P(C).25 and
  P(I).75.
• Note For the binomial model, we think of drawing
  of five chips with
• replacement from a box containing one C
  and three one Is. Each draw corresponds to one
  question and the probability of getting a C
  (correct answer) is always 0.25.

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Section 6.1 Sampling with Replacement

• Example 3 Roll a balanced die five times. Let
  Amore than one 6 and Bat least one 6.Find
  P(AB).
• Solution
• Consider a box filled with five chips
  marked N (not a 6) and one chip marked S (six).
  Now if we draw with replacement from this box, we
  get an S chip with probability 1/6. We must draw
  five chips, each draw corresponding to the roll
  of a balanced die.
• A2, 3, 4, or 5 S chips
• B1, 2, 3, 4, or 5 S chips
• So, A and B2, 3, 4, or 5 chips

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Section 6.1 Sampling with Replacement

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Section 6.3Waiting Time Model

• This section deals with a model which is
  similar to the binomial except that the number of
  draws is not fixed. The random variable of
  interest in this model is the number of trials
  needed for an event to occur. For this reason,
  we call it a waiting model.

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Section 6.3 Waiting Time Model

• Example 1 A coin has probability 2/3 of turning
  up head.
• The coin is tossed until four heads appear. Find
  the probability that six tosses are needed.
• The coin is tossed six times. Find the
  probability of getting four heads.
  

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Section 6.3 Waiting Time Model

• Solution

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Chapter 6. Waiting Time Model

• Class Exercises
• Do the following problems from your textbook
  (page 159)
• 1-6, 14.