05-Jan-2023 17:00:43

digital_dice_test():
  MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2
  Test digital_dice().

aandb():
  In game A, you flip a biased coin, which shows heads with probabiity]
  1/2 - epsilon; you win a dollar on heads.
  In game B, you have two biased coins.  If, at the time just before you
  decide to flip, your capital M is a multiple of 3 dollars, you chose
  coin 1, which shows heads with probability 1/10 - epsilon, otherwise
  you choose coin 2, which shows heads with probability 3/4 - epsilon.
  Both games A and B are losing games for you.
  But, paradoxically, if you randomly switch back and forth between
  one game and the other, you end up winning over the long term.

  Use graphics to display the winnings.
  Graphics saved as "aandb.png"

average():
  Use a Monte Carlo sample to estimate pi.

  Estimate for pi = 3.25381
  Error =           -0.112219
  Antithetic estimate for pi = 3.09509
  Error =                      0.046502

average():
  Use a Monte Carlo sample to estimate pi.

  Estimate for pi = 3.13568
  Error =           0.00591519
  Antithetic estimate for pi = 3.1412
  Error =                      0.000393039

average():
  Use a Monte Carlo sample to estimate pi.

  Estimate for pi = 3.14242
  Error =           -0.000824013
  Antithetic estimate for pi = 3.14162
  Error =                      -3.00045e-05

baby_boom():
  There are given probabilities of a man having 0, 1, 2, 3, 4, 5, 6 or 7 sons.
  What are the chances of having:
    2 sons in the second generation?
    4 sons in the second generation?
    6 sons in the third generation?

  Estimated probabilities:
  2 males in generation 2 = 0.0708
  4 males in generation 2 = 0.0423
  6 males in generation 3 = 0.0203

baby_boom():
  There are given probabilities of a man having 0, 1, 2, 3, 4, 5, 6 or 7 sons.
  What are the chances of having:
    2 sons in the second generation?
    4 sons in the second generation?
    6 sons in the third generation?

  Estimated probabilities:
  2 males in generation 2 = 0.06732
  4 males in generation 2 = 0.03964
  6 males in generation 3 = 0.02076

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.5
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 1.33487

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.5
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 1.99497

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.5
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 4.58118

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.5
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 5.15229

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.5
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 18.589

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.4
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 1.3859

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.4
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 2.08157

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.4
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 4.75358

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.4
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 5.3724

broke():
  Three players begin with L, M and N dollars.
  On each turn, each player flips a coin.
  All coins have the same bias P = 0.4
  If one player is "odd man out", he pays a dollar to each other player.
  When a player is bankrupt, the game is over.
  What is the average number of turns required?

  Average number of turns = 19.4407

bus():
  1 bus lines are available at a bus stop.
  In any hour, each bus line will come to the stop at a random time.
  A passenger arrives at the bus stop at a random time.
  What is the average wait for a bus?

  Estimated waiting time = 0.499813
  Theoretical time = 0.5

bus():
  2 bus lines are available at a bus stop.
  In any hour, each bus line will come to the stop at a random time.
  A passenger arrives at the bus stop at a random time.
  What is the average wait for a bus?

  Estimated waiting time = 0.333143
  Theoretical time = 0.333333

bus():
  3 bus lines are available at a bus stop.
  In any hour, each bus line will come to the stop at a random time.
  A passenger arrives at the bus stop at a random time.
  What is the average wait for a bus?

  Estimated waiting time = 0.250208
  Theoretical time = 0.25

bus():
  4 bus lines are available at a bus stop.
  In any hour, each bus line will come to the stop at a random time.
  A passenger arrives at the bus stop at a random time.
  What is the average wait for a bus?

  Estimated waiting time = 0.19995
  Theoretical time = 0.2

bus():
  5 bus lines are available at a bus stop.
  In any hour, each bus line will come to the stop at a random time.
  A passenger arrives at the bus stop at a random time.
  What is the average wait for a bus?

  Estimated waiting time = 0.166647
  Theoretical time = 0.166667

car():
  Park 3 cars in a line, and compute each car's nearest neighbor
  Estimate the probability that a given car is the nearest neighbor
  of its nearest neighbor.

  Estimated probability = 0.666667
  Theoretical probability is 0.666667

car():
  Park 10 cars in a line, and compute each car's nearest neighbor
  Estimate the probability that a given car is the nearest neighbor
  of its nearest neighbor.

  Estimated probability = 0.666675
  Theoretical probability is 0.666667

car():
  Park 20 cars in a line, and compute each car's nearest neighbor
  Estimate the probability that a given car is the nearest neighbor
  of its nearest neighbor.

  Estimated probability = 0.666727
  Theoretical probability is 0.666667

car():
  Park 30 cars in a line, and compute each car's nearest neighbor
  Estimate the probability that a given car is the nearest neighbor
  of its nearest neighbor.

  Estimated probability = 0.666598
  Theoretical probability is 0.666667

chess():
  A has to play three games of chess against B and C.
  A beats B with probability P, C with probability Q
  A has a choice of schedules: BCB or CBC.
  To win the tournament, A must win two successive games.
  Which schedule is better?

  Estimated win probability for BCB is = 0.792118
  Estimated win probability for CBC is = 0.863938

chess():
  A has to play three games of chess against B and C.
  A beats B with probability P, C with probability Q
  A has a choice of schedules: BCB or CBC.
  To win the tournament, A must win two successive games.
  Which schedule is better?

  Estimated win probability for BCB is = 0.395898
  Estimated win probability for CBC is = 0.575968

chess():
  A has to play three games of chess against B and C.
  A beats B with probability P, C with probability Q
  A has a choice of schedules: BCB or CBC.
  To win the tournament, A must win two successive games.
  Which schedule is better?

  Estimated win probability for BCB is = 0.192196
  Estimated win probability for CBC is = 0.203954

chess():
  A has to play three games of chess against B and C.
  A beats B with probability P, C with probability Q
  A has a choice of schedules: BCB or CBC.
  To win the tournament, A must win two successive games.
  Which schedule is better?

  Estimated win probability for BCB is = 0.064142
  Estimated win probability for CBC is = 0.07623

committee():
  From a faculty of 6 professors, 6 associate professors, 10
  assistant professors, and 12 instructors, a committee of size 6
  is formed randomly.  What is the probablity that there is at least
  one person of each rank in the committee?

  Estimated probability = 0.37798
  Exact probablity      = 0.379031

deli():
  A deli is open for 36,000 seconds.
  There are 1 or 2 clerks.
  Customers arrive at random.
  The amount of time it takes to service a customer is random.
  Customers are served in order.
  A clerk serves one customer at a time.
  Newly arrived customers wait in a queue if no clerk is available.
  What is the average and maximal times spent per customer?
  What is the average and maximal queue length?
  What percentage of time are the clerks idle?

  Average total time at deli 1014.34
  Maximum time at deli       3222
  Average queue length       8.59247
  Maximum queue length       29
  Percent idle time clerk1   6.58889
  Percent idle time clerk2   0

deli():
  A deli is open for 36,000 seconds.
  There are 1 or 2 clerks.
  Customers arrive at random.
  The amount of time it takes to service a customer is random.
  Customers are served in order.
  A clerk serves one customer at a time.
  Newly arrived customers wait in a queue if no clerk is available.
  What is the average and maximal times spent per customer?
  What is the average and maximal queue length?
  What percentage of time are the clerks idle?

  Average total time at deli 109.185
  Maximum time at deli       573
  Average queue length       0.117722
  Maximum queue length       3
  Percent idle time clerk1   52.3194
  Percent idle time clerk2   75.1222

deli():
  A deli is open for 36,000 seconds.
  There are 1 or 2 clerks.
  Customers arrive at random.
  The amount of time it takes to service a customer is random.
  Customers are served in order.
  A clerk serves one customer at a time.
  Newly arrived customers wait in a queue if no clerk is available.
  What is the average and maximal times spent per customer?
  What is the average and maximal queue length?
  What percentage of time are the clerks idle?

  Average total time at deli 9655
  Maximum time at deli       19882
  Average queue length       136.906
  Maximum queue length       264
  Percent idle time clerk1   0.125
  Percent idle time clerk2   0

deli():
  A deli is open for 36,000 seconds.
  There are 1 or 2 clerks.
  Customers arrive at random.
  The amount of time it takes to service a customer is random.
  Customers are served in order.
  A clerk serves one customer at a time.
  Newly arrived customers wait in a queue if no clerk is available.
  What is the average and maximal times spent per customer?
  What is the average and maximal queue length?
  What percentage of time are the clerks idle?

  Average total time at deli 1004.51
  Maximum time at deli       2465
  Average queue length       11.5885
  Maximum queue length       29
  Percent idle time clerk1   3.825
  Percent idle time clerk2   3.92222

dinner():
  N guests sit down at a dinner table, without noticing the name tags.
  What are the chances that no one sits at their assigned seat?

  N  P(derangement)

   1                     0
   2                   0.5
   3    0.3333333333333333
   4                 0.375
   5    0.3666666666666667
   6    0.3680555555555556
   7    0.3678571428571428
   8    0.3678819444444444
   9     0.367879188712522
  10    0.3678794642857143
  11    0.3678794392336059
  12    0.3678794413212816
  13    0.3678794411606912
  14    0.3678794411721619
  15    0.3678794411713971
  16     0.367879441171445
  17    0.3678794411714422
  18    0.3678794411714424
  19    0.3678794411714424
  20    0.3678794411714423

dish():
  Five dishwashers work together.  Five dishes are broken.
  What are the chances that at least four of the dishes
  are broken by the same particular dishwasher?

  Probability dishwasher #1 breaks at least 4 out of 5 dishes = 0.006734
  Theoretical probability is 0.00672

easywalk():
  A pedestrian begins 1000 blocks east and 1000 blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the exact expected wait for stop lights?

  Use graphics to display the expected results.
  Graphics saved as "easywalk.png"

election():
  N voters participate in an election.
  n of the voters are candidates.
  Voting rules may allow a voter to vote for themselves, or not.
  M votes are required in order for a leader to be chosen.
  What is the probability that, on a single vote, a leader will be chosen?

  Probability a leader was selected = 1

election():
  N voters participate in an election.
  n of the voters are candidates.
  Voting rules may allow a voter to vote for themselves, or not.
  M votes are required in order for a leader to be chosen.
  What is the probability that, on a single vote, a leader will be chosen?

  Probability a leader was selected = 0.77908

election():
  N voters participate in an election.
  n of the voters are candidates.
  Voting rules may allow a voter to vote for themselves, or not.
  M votes are required in order for a leader to be chosen.
  What is the probability that, on a single vote, a leader will be chosen?

  Probability a leader was selected = 0.0604

election():
  N voters participate in an election.
  n of the voters are candidates.
  Voting rules may allow a voter to vote for themselves, or not.
  M votes are required in order for a leader to be chosen.
  What is the probability that, on a single vote, a leader will be chosen?

  Probability a leader was selected = 0.10682

estimate():
  N runners participate in a marathon.
  Each runner wears a tag with their index, from 1 to N.
  We observe the values of K of these tags.
  We want to estimate N.

  Produce illustrative plots for several cases.
  Graphics saved as "estimate.png"

floss():
  A person buys two rolls of dental floss.
  Each roll has 40 feet of floss.
  The person randomly selects a roll and takes 1 foot of floss.
  When one roll runs out, how many feet remain in the other roll?

  Average remaining floss = 7.11423

floss():
  A person buys two rolls of dental floss.
  Each roll has 150 feet of floss.
  The person randomly selects a roll and takes 1 foot of floss.
  When one roll runs out, how many feet remain in the other roll?

  Average remaining floss = 13.8083

forgetful_burglar():
  In a town of 201 homes, a burglar starts at home 101.
  He randomly moves one or two homes left or right.
  What is the typical number of moves he will make before revisiting a home?

   K    Prob(K)

   1  0
   2  0.250171
   3  0.281204
   4  0.195319
   5  0.117215
   6  0.0683627
   7  0.0388426
   8  0.0217806
   9  0.0121527
  10  0.0066848
  Graphics saved as "fb.png"

gameb():
  In game B, you have two biased coins.  If, at the time just before you
  decide to flip, your capital M is a multiple of 3 dollars, you chose
  coin 1, which shows heads with probability 1/10 - epsilon, otherwise
  you choose coin 2, which shows heads with probability 3/4 - epsilon.
  Game B is a losing game for you, and this code simply demonstrates
  that using many simulations.

  Produce a plot showing how the player loses.
  Graphics saved as "gameb.png"

gs():
  A building has 7 floors, and there are n elevators, each of which
  is at a randomly chosen floor.
  A person on floor 2 requests an elevator, wishing to go up.
  What is the probability that the first elevator to arrive is going down?

  Estimated probability of down elevator = 0.833097
  Theoretical probability is 0.833333

gs():
  A building has 7 floors, and there are n elevators, each of which
  is at a randomly chosen floor.
  A person on floor 2 requests an elevator, wishing to go up.
  What is the probability that the first elevator to arrive is going down?

  Estimated probability of down elevator = 0.722038
  Theoretical probability is 0.722222

gs():
  A building has 7 floors, and there are n elevators, each of which
  is at a randomly chosen floor.
  A person on floor 2 requests an elevator, wishing to go up.
  What is the probability that the first elevator to arrive is going down?

  Estimated probability of down elevator = 0.647621
  Theoretical probability is 0.648148

guess():
  Given M items of ranks 1 through M,
  randomly guess the rank of each item. 
  On average, how many ranks will we guess correctly?

  Average number of correct pairings = 1.0005
  Expected value is 1.

jury():
  There are 5 judges on an appeals court.
  Each judge has a probability of making a correct ruling.
  What is the probability that a majority of the judges will rule incorrectly?

  Probability of a mistaken judgement = 0.0070078

kelvin():
  A biased coin comes up heads with probability 0.4.
  To get a unbiased random value, toss the coin twice.
  If you get TH, call it heads; if you get HT, call it tails.
  If you get TT or HH, do another double toss.
  On average, how many double tosses are necessary?

  Average number of double tosses = 2.0761
  Theoretical value               = 2.08333

malt():
  Lil and Bill agree to meet in the malt shop between 3:30 and 4:00.
  Each arrives at a random time.
  Lil will wait 5 minutes, then leave.
  Bill will wait 7 minutes, then leave.
  What is the probability of a meeting?

  Estimated meeting probability = 0.358859
  Theoretical probability is 0.358889

missing():
  There are 100 senators.
  A bill needs a majority of present senators to pass.
  A senators are against the bill.
  M senators are missing the vote.
  What is the probability that the bill will be defeated?

  Probability of defeat = 0.128476

missing():
  There are 100 senators.
  A bill needs a majority of present senators to pass.
  A senators are against the bill.
  M senators are missing the vote.
  What is the probability that the bill will be defeated?

  Probability of defeat = 0.063577

missing():
  There are 100 senators.
  A bill needs a majority of present senators to pass.
  A senators are against the bill.
  M senators are missing the vote.
  What is the probability that the bill will be defeated?

  Probability of defeat = 0.193682

mono():
  Expected value of the number of random numbers that
  can be generated, which are monotone increasing.

  Extimated expected length = 2.71805
  Expected value is 2.71828.

obtuse():
  Define a "random" triangle as half of a rectangle
  with height 1 and width 1 <= L.
  What are the chances the triangle is obtuse?
  Using value L = 1

  Estimated likelihoood of obtuse triangle = 0.725212
  For L=1, expected value = 0.725206
  For L=2, expected value = 0.798374

obtuse1():
  Define a "random" triangle by splitting the unit
  interval into three random pieces.
  What are the chances the triangle is obtuse?

  Estimated likelihoood of obtuse triangle = 0.17034
  Exact value is 0.170558

optimal():
  A dating club offers 11 potential partners.
  It turns out that any of 2 of these partners would be acceptable.
  The dater gets 1 date with each partner, but immediately after the
  date, must either marry that partner, or move to the next date.
  The dater plans to date a sample of the partners without a marriage
  offer, and then marry the next partner who is better than all
  the sample dates. 
  As the sample size is varied, what are the chances of happiness?

  Use graphics to display result.
  Graphics saved as "optimal.png"

optimal():
  A dating club offers 50 potential partners.
  It turns out that any of 5 of these partners would be acceptable.
  The dater gets 1 date with each partner, but immediately after the
  date, must either marry that partner, or move to the next date.
  The dater plans to date a sample of the partners without a marriage
  offer, and then marry the next partner who is better than all
  the sample dates. 
  As the sample size is varied, what are the chances of happiness?

  Use graphics to display result.
  Graphics saved as "optimal.png"

patrol():
  Consider a divided highway, and suppose it could be divided by
  a grassy median or a concrete barrier.
  Suppose NUMBER police cars patrol the highway.
  Suppose an accident occurs at a randomly chosen location and lane.
  Suppose all patrol cars immediately head towards the accident.
  If a grassy median, then patrol cars in the wrong lane can 
  immediately reverse direction.
  For the concrete barrier, patrol cars in the wrong lane must
  continue to the end of the highway and then turn around.
  Estimate the average time required by a patrol car to reach the accident.

  Average grass distance    = 0.333548
  Average concrete distance = 0.999518

patrol():
  Consider a divided highway, and suppose it could be divided by
  a grassy median or a concrete barrier.
  Suppose NUMBER police cars patrol the highway.
  Suppose an accident occurs at a randomly chosen location and lane.
  Suppose all patrol cars immediately head towards the accident.
  If a grassy median, then patrol cars in the wrong lane can 
  immediately reverse direction.
  For the concrete barrier, patrol cars in the wrong lane must
  continue to the end of the highway and then turn around.
  Estimate the average time required by a patrol car to reach the accident.

  Average grass distance    = 0.208144
  Average concrete distance = 0.666751

patrol():
  Consider a divided highway, and suppose it could be divided by
  a grassy median or a concrete barrier.
  Suppose NUMBER police cars patrol the highway.
  Suppose an accident occurs at a randomly chosen location and lane.
  Suppose all patrol cars immediately head towards the accident.
  If a grassy median, then patrol cars in the wrong lane can 
  immediately reverse direction.
  For the concrete barrier, patrol cars in the wrong lane must
  continue to the end of the highway and then turn around.
  Estimate the average time required by a patrol car to reach the accident.

  Average grass distance    = 0.149825
  Average concrete distance = 0.500455

patrol():
  Consider a divided highway, and suppose it could be divided by
  a grassy median or a concrete barrier.
  Suppose NUMBER police cars patrol the highway.
  Suppose an accident occurs at a randomly chosen location and lane.
  Suppose all patrol cars immediately head towards the accident.
  If a grassy median, then patrol cars in the wrong lane can 
  immediately reverse direction.
  For the concrete barrier, patrol cars in the wrong lane must
  continue to the end of the highway and then turn around.
  Estimate the average time required by a patrol car to reach the accident.

  Average grass distance    = 0.116791
  Average concrete distance = 0.399881

patrol():
  Consider a divided highway, and suppose it could be divided by
  a grassy median or a concrete barrier.
  Suppose NUMBER police cars patrol the highway.
  Suppose an accident occurs at a randomly chosen location and lane.
  Suppose all patrol cars immediately head towards the accident.
  If a grassy median, then patrol cars in the wrong lane can 
  immediately reverse direction.
  For the concrete barrier, patrol cars in the wrong lane must
  continue to the end of the highway and then turn around.
  Estimate the average time required by a patrol car to reach the accident.

  Average grass distance    = 0.095128
  Average concrete distance = 0.333222

pierror():
  Estimate pi by counting random points in the unit
  square which are also in the quarter circle.

  Using 100 points, estimate is 0.89, absolute error 2.25159

pierror():
  Estimate pi by counting random points in the unit
  square which are also in the quarter circle.

  Using 10000 points, estimate is 0.7819, absolute error 2.35969

ranking():
  A list of 24 items is given.
  The test taker is required to give a rank for each.
  For each item, the test taker randomly chooses a value between 1 and M.
  What is the average number of correct rankings?

  Average number of correct matches = 0.998299

rhs():
  Random Harmonic Series:
  Compute and histogram many values of the partial sums of 
    sum ( 1 <= k < infinity ) t(i) / k
  where t(i) is randomly +1 or -1.
  Graphics saved as "rhs.png"

rolls():
  Two rolls of toilet paper are installed in a toilet, with 200 sheets.
  There are two kinds of people, with probabilities p and 1-p.
  * big choosers take one sheet from the larger roll;
  * little choosers take one sheet from the smaller roll (unless empty).
  When one roll becomes empty, how many sheets are on the other roll?
  Use graphics to display results.
  Graphics saved as "rolls.png"

smoker():
  A smoker buys two packs of 40 matches.
  He then repeatedly selects a match from a randomly chosen pack.
  When one pack runs out, how many matches have been used in total?

  Average total number of matches used = 72.886
  Graphics saved as "smoker.png"

smokerb():
  A smoker buys two packs of 40 matches.
  He then repeatedly selects a match from a randomly chosen pack.
  At some point, the pack he chooses will be empty.
  How many matches have been used by then?

  Average total number of matches used = 73.795
  Graphics saved as "smokerb.png"

spin():
  A game involves two spinnable disks, each divided into three sectors.
  A player spins disk 1 or 2 according to the following rules:
  * if the player spins disk i, and it stops in region Pij, he
    moves from disk i to disk j;
  * if the spinner stops in region Pi3, the game ends.
  * if the game ends in P13, the player wins.
  What is the probablity that the player, starting with disk 1, wins?

  Probabiity of winning = 0.6596
  Theoretical value     = 0.65

steve():
  Steve gets on an elevator going up.  There are 11 higher floors.
  Steve wishes to go up 9 floors.
  There are K additional riders in the elevator, each of whom has
  randomly chosen one of the 11 higher floors as destination.
  On average, how many times will the elevator stop until Steve
  reaches his floor?

  Estimated number of stops = 1
  Theoretical number is 1

steve():
  Steve gets on an elevator going up.  There are 11 higher floors.
  Steve wishes to go up 9 floors.
  There are K additional riders in the elevator, each of whom has
  randomly chosen one of the 11 higher floors as destination.
  On average, how many times will the elevator stop until Steve
  reaches his floor?

  Estimated number of stops = 1.72675
  Theoretical number is 1.72727

steve():
  Steve gets on an elevator going up.  There are 11 higher floors.
  Steve wishes to go up 9 floors.
  There are K additional riders in the elevator, each of whom has
  randomly chosen one of the 11 higher floors as destination.
  On average, how many times will the elevator stop until Steve
  reaches his floor?

  Estimated number of stops = 2.38875
  Theoretical number is 2.38843

steve():
  Steve gets on an elevator going up.  There are 11 higher floors.
  Steve wishes to go up 9 floors.
  There are K additional riders in the elevator, each of whom has
  randomly chosen one of the 11 higher floors as destination.
  On average, how many times will the elevator stop until Steve
  reaches his floor?

  Estimated number of stops = 2.9886
  Theoretical number is 2.98948

steve():
  Steve gets on an elevator going up.  There are 11 higher floors.
  Steve wishes to go up 9 floors.
  There are K additional riders in the elevator, each of whom has
  randomly chosen one of the 11 higher floors as destination.
  On average, how many times will the elevator stop until Steve
  reaches his floor?

  Estimated number of stops = 5.60598
  Theoretical number is 5.60722

stopping():
  From a population of 5 indexed values, the highest is sought.
  Values are to be discovered in order of index.
  When value I is discovered:
  * it may be rejected, and the next value discovered, or
  * it may be accepted, and the process is terminated.
  A strategy is to view S items in a row, and then accept 
  the very next item that is larger than max(S).
  Given N, what is S, and what are the chances that this process will
  produce the maximum?

  Population size N      = 5
  Optimal sample size S  = 2
  Ratio N / S            = 2.5
  Probability of success = 0.433333

stopping():
  From a population of 10 indexed values, the highest is sought.
  Values are to be discovered in order of index.
  When value I is discovered:
  * it may be rejected, and the next value discovered, or
  * it may be accepted, and the process is terminated.
  A strategy is to view S items in a row, and then accept 
  the very next item that is larger than max(S).
  Given N, what is S, and what are the chances that this process will
  produce the maximum?

  Population size N      = 10
  Optimal sample size S  = 3
  Ratio N / S            = 3.33333
  Probability of success = 0.39869

stopping():
  From a population of 20 indexed values, the highest is sought.
  Values are to be discovered in order of index.
  When value I is discovered:
  * it may be rejected, and the next value discovered, or
  * it may be accepted, and the process is terminated.
  A strategy is to view S items in a row, and then accept 
  the very next item that is larger than max(S).
  Given N, what is S, and what are the chances that this process will
  produce the maximum?

  Population size N      = 20
  Optimal sample size S  = 7
  Ratio N / S            = 2.85714
  Probability of success = 0.384209

stopping():
  From a population of 50 indexed values, the highest is sought.
  Values are to be discovered in order of index.
  When value I is discovered:
  * it may be rejected, and the next value discovered, or
  * it may be accepted, and the process is terminated.
  A strategy is to view S items in a row, and then accept 
  the very next item that is larger than max(S).
  Given N, what is S, and what are the chances that this process will
  produce the maximum?

  Population size N      = 50
  Optimal sample size S  = 18
  Ratio N / S            = 2.77778
  Probability of success = 0.374275

stopping():
  From a population of 100 indexed values, the highest is sought.
  Values are to be discovered in order of index.
  When value I is discovered:
  * it may be rejected, and the next value discovered, or
  * it may be accepted, and the process is terminated.
  A strategy is to view S items in a row, and then accept 
  the very next item that is larger than max(S).
  Given N, what is S, and what are the chances that this process will
  produce the maximum?

  Population size N      = 100
  Optimal sample size S  = 37
  Ratio N / S            = 2.7027
  Probability of success = 0.371043

sylvester_quadrilateral():
  Estimate the probability that four points, chosen
  uniformly at random in the unit circle, form a 
  concave (=nonconvex) quadrilateral.

  Estimated concave probability = 0.295555
  Expected value is 0.29552.

umbrella():
  A person has XI umbrellas at home, and YI at the office.
  With probability P, it will be raining at any given time.
  If it is raining, the person takes an umbrella from one place to the other.
  How many walks will the person take before running out of umbrellas?
  Use graphics to display results.
  Graphics saved as "umbrella.png"

umbrella():
  A person has XI umbrellas at home, and YI at the office.
  With probability P, it will be raining at any given time.
  If it is raining, the person takes an umbrella from one place to the other.
  How many walks will the person take before running out of umbrellas?
  Use graphics to display results.
  Graphics saved as "umbrella.png"

walk():
  A pedestrian begins M blocks east and M blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the average wait for stop lights?

  Estimated waiting time = 0.75411

walk():
  A pedestrian begins M blocks east and M blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the average wait for stop lights?

  Estimated waiting time = 1.22616

walk():
  A pedestrian begins M blocks east and M blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the average wait for stop lights?

  Estimated waiting time = 1.76292

walk():
  A pedestrian begins M blocks east and M blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the average wait for stop lights?

  Estimated waiting time = 2.50072

walk():
  A pedestrian begins M blocks east and M blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the average wait for stop lights?

  Estimated waiting time = 3.95794

walk():
  A pedestrian begins M blocks east and M blocks north of a destination.
  At each intersection, there is a stop light which is set randomly,
  and switches after 1 minute.
  Until reaching avenue 1 or street 1, the pedestrian always crosses
  the intersection in accordance with the stop light.
  Thereafter, the pedestrian must wait at each stop light encountered.
  What is the average wait for stop lights?

  Estimated waiting time = 5.63146

digital_dice_test():
  Normal end of execution.

05-Jan-2023 17:02:00