In a class of 80 students, the professor calls on 1 student chosen at random for a recitation in each class period. There are 32 class periods in a term.
- Write a formula for the exact probability that a given student is called upon [math]j[/math] times during the term.
- Write a formula for the Poisson approximation for this probability. Using your formula estimate the probability that a given student is called upon more than twice.
Assume that we are making raisin cookies. We put a box of 600 raisins into our dough mix, mix up the dough, then make from the dough 500 cookies. We then ask for the probability that a randomly chosen cookie will have 0, 1, 2, ... raisins. Consider the cookies as trials in an experiment, and let [math]X[/math] be the random variable which gives the number of raisins in a given cookie. Then we can regard the number of raisins in a cookie as the result of [math]n = 600[/math] independent trials with probability [math]p = 1/500[/math] for success on each trial. Since [math]n[/math] is large and [math]p[/math] is small, we can use the Poisson approximation with [math]\lambda = 600(1/500) = 1.2[/math]. Determine the probability that a given cookie will have at least five raisins.
For a certain experiment, the Poisson distribution with parameter [math]\lambda = m[/math] has been assigned. Show that a most probable outcome for the experiment is the integer value [math]k[/math] such that [math]m - 1 \leq k \leq m[/math]. Under what conditions will there be two most probable values? Hint: Consider the ratio of successive probabilities.
When John Kemeny was chair of the Mathematics Department at Dartmouth College, he received an average of ten letters each day. On a certain weekday he received no mail and wondered if it was a holiday. To decide this he computed the probability that, in ten years, he would have at least 1 day without any mail. He assumed that the number of letters he received on a given day has a Poisson distribution. What probability did he find? Hint: Apply the Poisson distribution twice. First, to find the probability that, in 3000 days, he will have at least 1 day without mail, assuming each year has about 300 days on which mail is delivered.
Reese Prosser never puts money in a 10-cent parking meter in Hanover. He assumes that there is a probability of .05 that he will be caught. The first offense costs nothing, the second costs 2 dollars, and subsequent offenses cost 5 dollars each. Under his assumptions, how does the expected cost of parking 100 times without paying the meter compare with the cost of paying the meter each time?
Feller[Notes 1] discusses the statistics of flying bomb hits in an area in the south of London during the Second World War. The area in question was divided into [math]24 \times 24 = 576[/math] small areas. The total number of hits was 537. There were 229 squares with 0 hits, 211 with 1 hit, 93 with 2 hits, 35 with 3 hits, 7 with 4 hits, and 1 with 5 or more. Assuming the hits were purely random, use the Poisson approximation to find the probability that a particular square would have exactly [math]k[/math] hits. Compute the expected number of squares that would have 0, 1, 2, 3, 4, and 5 or more hits and compare this with the observed results.
Notes
Assume that the probability that there is a significant accident in a nuclear power plant during one year's time is .001. If a country has 100 nuclear plants, estimate the probability that there is at least one such accident during a given year.
An airline finds that 4 percent of the passengers that make reservations on a particular flight will not show up. Consequently, their policy is to sell 100 reserved seats on a plane that has only 98 seats. Find the probability that every person who shows up for the flight will find a seat available.
The king's coinmaster boxes his coins 500 to a box and puts 1 counterfeit coin in each box. The king is suspicious, but, instead of testing all the coins in 1 box, he tests 1 coin chosen at random out of each of 500 boxes. What is the probability that he finds at least one fake? What is it if the king tests 2 coins from each of 250 boxes?