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Category Archives: approximation of events
Overloading an Elevator
A new elevator in a large hotel is designed to carry about 30 people, with a total weight of up to 5000 lbs. More that 5000 lbs. overloads the elevator. The average weight of guests at the hotel is 150 lbs., with a standard deviation of 55 lbs. Suppose 30 of the hotel’s guests get into the elevator . Assuming the weights of the guests are independent random variables, what is the chance of overloading the elevator ? Give your approximate answer as a decimal.
[Pitman p 204, # 19]
Boxes without toys
A cereal company advertises a prize in every box of its cereal. In fact, only about 95% of the boxes have a prize in them. If a family buys one box of this cereal every week for a year, estimate the chance that they will collect more than 45 prizes. What assumptions are you making ?
[Pitman p122, # 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.
- 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]
Lottery
Suppose that each week you buy a ticket in a lottery which has a chance \(\frac1{100}\) of wining. If you do this every week of a year, approximately what is the chance of getting exactly \(k\) wins for \(k=0,1,2,3\).
[Pittman p122, # 5]
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]
Birthday problem
The birthday problem is a classic problem in probability.
Given \(n\) people in a room what is the probability that at least two of them have the same birthday ?
- Compute \(\mathbf{P}(n)\) assuming that a person being born on any day is equal.
In blogs Andy Gelman and Chris Mulligan talk about how the uniformity assumption may be incorrect and the effect this has on the birthday problem.
- Chris examined the uniformity assumption by looking at CDC data for one year in terms of number of births. He provides R code (that I slightly adapted) that you can run in RStudio to plot the number of births through the year. How different is this from uniform ?
- Given this observed distribution he then computes the difference between the result of the birthday problem given the observed distribution versus a uniform distribution. This is done using Monte Carlo simulation in R (again slightly adapted by me). Does the deviation from the uniform distribution have a strong effect ?
Inclusion of origin
Draw \(n\) points from the uniform distribution on the circle and draw the convex hull around these points. What is the probability that the origin (center of the circle) is contained in the convex hull ?
[From: The Probabilistic Method by Alon and Spencer]
The matching problem
There are \(n\) letters addressed to \(n\) eople at different addresses. The \(n\) addresses are typed on \(n\) envelopes. A disgruntled secretary shuffles the letters and puts them in the envelopes in random order, one letter per envelope.
- Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.]
- What is the probability approximately, for large \(n\) ?
[ For example, the needed inclusion-exclusion formula is given in Problem 12, p. 31 in Pitman]
You will also need to know the number of elements in the set
\[ \{ (i_1,i_2,\cdots, i_k) : 1 \leq i_1 < i_2 < \cdots < i_k\leq n\} \]
which is discussed here.
Looking for a rare couple
A certain advertising firm needs to find a recently married couple who are both born on April 19th. They send the intern to the city hall in NYC to look through the records.
- If she looks through 50,000 records, what is the chance that she finds at least one such couple?
- If you were to approximate the above probability with a limit theorem what distribution would you use? Explain why.
- If she relaxes her search criteria to only look for couples with the same birthday, what is the chance she finds at least one such couple after looking through 50,000 records?
- If you were to approximate the above probability with a limit theorem what distribution would you use? Explain why.
- Before starting, the intern is interested in estimating how long this task might take. How many records must she consider to have a 80% chance of finding a couple matching the first search criteria? What about the second search criteria?
In the above, assume that the date people are born is uniform over the year and each person’s birthday is independent of each other person’s birthday. Also, assume that the day one is born does not influence one’s choice of spouse.
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.
