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Introduction |
An Empirical Probability
|
The Game of Sorry!
|
Exercises
Introduction
A probability may be a very complicated number to compute. It is reasonably easy to use theoretical methods when computing probabilities involving decks of cards and coin tosses because you can often list or easily count all the possible ways a desired outcome might occur.
However if you pose a question such as “What is the probability of an auto accident?” it becomes impossible to consider all the factors involved. A theoretical analysis of this question would involve consideration of every way an auto accident could occur (already a daunting task), knowledge of the driving skills of every driver, the driving patterns of all drivers and the chances a bad or careless driver would encounter another driver at precisely the right instant to cause an accident. Techniques involving empirical data must be used.
Even in cases where theoretical methods may be applied, the process can be tricky. Something as seemingly simple as a common board game may present a very difficult problem. A few years back, several mathematically- and statistically-minded people attempted to study the game of Monopoly from a theoretical standpoint. Their work can be found at Web sites such as
Monopoly Revisited by Ian Stewart
This project looks at both types of probability computation, empirical and theoretical.
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An Empirical Probability
If a coin is a fair coin, then you would expect the number of heads to be roughly half the number of times you tossed it. Here you are using the theoretical probability of ½ for a head. If you suspected a coin was slightly top heavy how would you compute the probability of a head? You have no theory at your disposal.
The answer is, you would flip the coin a lot (emphasis on “a lot”) and count the number of heads. If you flipped the coin 1,000 times and saw 650 heads, you would estimate the probability of a head to be 650/1000 = 0.65 . This might not be the exact probability. Flipping 10,000 times might reveal 6486 heads or a probability of 0.6486 but you would be confident your estimate was close.
This is how probabilities are often computed for events far too complicated to analyze theoretically. For example, consider a health-related probability. The National Center for Health Statistics, a division of the Center for Disease Control, (
http://www.cdc.gov/nchs
) collects and publishes data of a health related nature. The web site mostly contains data that have been already statistically summarized along with a variety of factoids. The “FASTATS A to Z” link off the main page leads to an alphabetical list of topics that are interesting to peruse.
For instance, a look at the summary health statistics under Allergies reveals the fact that in 2003, there were 18.356 million cases of hay fever in the US among people 18 years of age or older, in other words, among adults. How would we convert this to a probability?
To compute the probability of hay fever, we would need to divide the number of observed cases with the number of possible cases, i.e. the population. Note that our hay fever data refers specifically to people of a certain age group not the entire population. Therefore we need the adult U.S. population in 2003, since this is the year the hay fever number was collected. A visit to the Census bureau web site (
http://www.census.gov/
) provides us with an estimate of 217.738 million adults in the U.S. in 2003. Thus we estimate
Probability of hay fever:
.
Note that due to the large amount of time needed to accurately collect such health and census data, we often have to use data from a number of years prior to the current year in these estimates. The exercises will have you seek out similar data to estimate probabilities.
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The Game of Sorry!
Sorry! is a long-popular board game produced by Parker Brothers, the makers of Monopoly, now part of Hasbro, Inc. The game is rather easy to play but still there are interesting probability computations lurking therein.
A schematic diagram of the Sorry! game board appears below.
We’ll give a quick summary of the rules. More information about the game and its manufacturer can be found at http://www.hasbro.com. A detailed set of rules is available at
http://www.hasbro.com/common/instruct/Sorry.PDF
.
In Sorry!, you (one of the players) first choose a color for your play pieces (called pawns) and place the corresponding four pawns in your color’s Start position (labeled S in the above diagram.) The object of the game is to move all four of your pawns out of the starting box, then to travel clockwise around the board until all 4 pawns are in your color’s home space, marked H in the diagram. At any point during the game, any number of your pawns can be out of Start and circling the board. The first player to lead all four of their pawns home is the winner.
Moves are determined from cards drawn from a special deck of cards shuffled and placed face down in the center of the board. Players take turns drawing cards, following the instructions that appear on their drawn card. Most cards contain a number but a player is often given a choice of how to apply that number. Here’s a summary of the numbered cards and their meanings:
1. Move one pawn out of start onto the square in front of Start or move any pawn 1 space.
2. Move one pawn out of start onto the square in front of Start or move any pawn other pawn 2 spaces. After choosing which move you wish to take, you are allowed to draw another card on the same turn. Note that if you choose the first option, the pawn is simply allowed to leave start, it does not move 2 spaces.
3. Move one pawn 3 spaces.
4. Move one pawn backward (counterclockwise) 4 spaces.
5. Move one pawn forward 5 spaces.
7. A move of 7 forward or the 7 may be split to move two different pawns a total of 7 spaces.
8. Move one pawn 8 spaces.
10. Move one pawn either 10 spaces forward or 1 space backward.
11. Move one pawn 11 spaces or swap places with any pawn of a different color.
12. Move one pawn forward 12 spaces.
Note the only way for a pawn to leave the starting gate is by drawing a 1 or 2. For a pawn to enter Home, a move must land the pawn exactly in Home. Thus if you need to move your last pawn 5 spaces to win the game and you draw an 8, you are out of luck. If ever during the game, you draw a card and you cannot perform the indicated move, you lose your turn. When the deck is exhausted, it is reshuffled and placed face down in the center of the board and play continues.
Now with just these rules, the game of Sorry! would be just a race around the board. To make things more interesting, there are several ways you can set an opponent back by sending one of their pawns back to their Start.
· If at the end of your move, your pawn lands on a space occupied by an opponent’s pawn, the opponent’s pawn returns to its Start.
· If you draw one of the special cards marked Sorry! you may take a pawn from your Start and place it on any space occupied by an opponent’s pawn, sending that pawn back to its start.
· Each side of the board contains slides, indicated by the arrows in the above diagram. If at the end of your move, your pawn occupies the space containing the tail of a slide not of your pawn’s color, your pawn moves the length of the slide sending any opponent’s pawns occupying the slide back to Start.
The five colored squares leading up to your Home are safe spaces. A pawn occupying one of these spaces cannot be sent back to Start.
To compute probabilities related to Sorry! you need to know the distribution of cards in the deck. There are 45 cards in the deck, four each of the 2, 3, 4, 5, 7, 8, 10, 11, 12 and Sorry! cards and five of the 1 card.
In a game of Sorry! you are the color blue and you have one opponent who is color red. At one point in the game you have exhausted the deck so it has been reshuffled. The board at that point looks like
and it is your turn. What is the probability that you will be able to get a pawn Home on your turn?
First of all, the only pawn you could possibly get Home on a single turn is the one nearest Home,
so you focus your attention on moving that pawn, which is 10 spaces from Home. So that’s one possibility right there. If you draw a 10 (an event having probability 4/45) the pawn is Home. Are there other ways?
Don’t forget if you draw a 2, you may move two spaces and then draw again, it is still your turn. Thus another way to bring the pawn Home is by drawing a 2 and then an 8, an event with probability
.
How about if you draw 2 and then another 2? You move the pawn four spaces closer to Home leaving six spaces to go. There is no card which moves you exactly 6 spaces but you could draw a 7 allowing you to move this pawn 6 spaces and one other pawn 1 space because a 7 may be split. Thus 2-2-7 is a possibility and similarly 2-2-2-7 and 2-2-2-2-7. These events have probabilities of
, and
respectively.
All told the probability of sending this pawn Home on your turn is
or roughly a 1 in 10 chance.
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Exercises
When you are finished reviewing this Project, go on to the exercises below.
1. Using data from 2003, estimate the probability that a randomly selected adult person in the U.S. will be afflicted with asthma. Use this probability to estimate the number of Americans afflicted in 2004.
2. A Web Search Exercise Using data from any year in the last 10 years, estimate the probability a newborn baby will be female. Locate the necessary data on the World Wide Web and submit the relevant URLs along with your answer.
3. In the Sorry! game pictured
assume the deck has just been reshuffled and it is blue’s turn. Compute the probability that blue will be able to send one of red’s pawns back to Start on this turn. Be careful to think about all the ways this could happen. For example, with a single card, the blue pawn in the picture below can send the red pawn back.
4. Again assume the same configuration as in Exercise 3 but this time it is red’s turn and the cards have not just been reshuffled. In fact, you blue, have been watching the cards being drawn and you know the following cards are gone from the deck
Card
1
2
3
4
5
7
8
10
11
12
Sorry!
# already drawn
1
4
1
1
0
2
1
2
3
0
2
Compute the probability that red will send one of your men back to Start on this turn.
