# Tag Archives: JCM_math230_HW3_S13

## Defective Machines

Suppose that the probability that an item produced by a certain machine will be defective is 0.12.

1.  Find the probability (exactly)  that a sample of 10 items will contain at most 1 defective item.
2. 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.

1. find $$m$$
2. find the probability that  $$\mathbf{P}(S=m)$$ correct to 3 decimal places.
3. what is the normal approximation to $$\mathbf{P}(S=m)$$  ?
4. what is the Poisson approximation to $$\mathbf{P}(S=m)$$ ?
5. 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 ?
6. 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.

1. Assuming that passengers show up independently of each other, what is the chance that the flight will be overbooked ?
2. Suppose that people tend to travel in groups. Would that increase of decrease the probability of overbooking ? Explain your answer.
3. 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.

1. Without calculations, say which of these outcomes is more likely. Why ?

[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.

1. 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?
2. 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?