Tag Archives: JCM_math340_HW2_F13

Finding a good phone

At the London station there are three pay phones which accept 20p coins. one never works, another works, while the third works with probability 1/2. On my way to London for the day, I wish to identify the reliable phone, so that I can use it on my return. The station is empty and I have just three 20p coins. I try one phone and it doesn’t work. I try another twice in succession and it works both times. What is the probability that this second phone is the reliable one ?

 

 

[Suhov and Kelbert, p.10, problem 1.9]

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]

Polya’s urn

An urn contains \(4\) white balls and \(6\) black balls. A ball is chosen at random, and its color is noted. The ball is then replaced, along with \(3\) more balls of the same color. Then another ball is drawn at random from the urn.

  1. Find the chance that the second ball drawn is white.
  2. Given the second ball drawn is white, what is the probability that the first ball drawn is black ?
  3. Suppose the original contents of the urn are \(w\) white and \(b\) black balls. Also after drawing a ball we replace with \(d\) balls of the same color. What is the probability that the second ball drawn is white (it should be \(\frac{w}{w+b}\) )?

[Pitman page 53. Problem 2]

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]

The chance of being English

English and American spellings are rigour and rigor, respectively. An English speaking guest staying at a Paris hotel writes the word and chose a letter at random from his spelling. The letter turns out to be a vowel. (that is any of : e,a,i,o,u). If 40% of the English speaking guests are American and 60% are English, what is the probability that the writer is American ?

 

 

 

[Ross, p. 107 #29]

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 ?
  2. Confirm your answer to the previous question by a calculation.

 

[Pitman, p. 108 #3]

 

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.