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Defective Machines
Suppose that the probability that an item produced by a certain machine will be defective is 0.12.
- Find the probability (exactly) that a sample of 10 items will contain at most 1 defective item.
- Use the Poisson to approximate the preceding probability. Compare your two answers.
[Inspired Ross, p. 151, example 7b ]
Approximation: Rare vs Typical
Let \(S\) be the number of successes in 25 independent trials with probability \(\frac1{10}\) of success on each trial. Let \(m\) be the most likely value of S.
- find \(m\)
- find the probability that \(\mathbf{P}(S=m)\) correct to 3 decimal places.
- what is the normal approximation to \(\mathbf{P}(S=m)\) ?
- what is the Poisson approximation to \(\mathbf{P}(S=m)\) ?
- repeat the first part of the question with the number of trial equal to 2500 rather than 25. Would the normal or Poisson approximation give a better approximation in this case ?
- repeat the first part of the question with the number of trial equal to 2500 rather than 25 and the probability of success as \(\frac1{1000}\) rather that \(\frac1{10}\) . Would the normal or Poisson approximation give a better approximation in this case ?
[Pitman p122 # 7]
Airline Overbooking
An airline knows that over the long run, 90% of passengers who reserve seats for a flight show up. On a particular flight with 300 seats, the airline sold 324 reservations.
- Assuming that passengers show up independently of each other, what is the chance that the flight will be overbooked ?
- Suppose that people tend to travel in groups. Would that increase of decrease the probability of overbooking ? Explain your answer.
- Redo the the calculation in the first question assuming that passengers always travel in pairs. Are your answers to all three questions consistent ?
[Pitman p. 109, #9]
Coin Flips: typical behavior
A fair coin is tossed repeatedly. Considering the following two possible outcomes:
55 or more heads in the first 100 tosses.
220 or more heads in the first 400 tosses.
- Without calculations, say which of these outcomes is more likely. Why ?
- Confirm your answer to the previous question by a calculation.
[Pitman, p. 108 #3]
Leukemia Test
A new drug for leukemia works 25% of the time in patients 55 and older, and 50% of the time
in patients younger than 55. A test group has 17 patients 55 and older and 12 patients younger than 55.
- A uniformly random patient is chosen from the test group, and the drug is administered and it is a success. What is the probability the patient was 55 and older?
- A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all of them?
School admissions
The ideal size of the freshman class of a small southern college is 150 students. Given that a student is admitted to the school, historical data indicates the student will actually attend with a probability 0.3. (We will assume that students make decisions independently of each other, even though this is certainly not true in reality). Approximately what is the chance that more than 150 students accept if 450 students are admitted.
