We use probability to model reproducible experiments. An experiment yields observations. An event is a set of observations. A probability model tells us the probabilities of events.
This tutorial discusses a good way to think of probability models. They work like this: you write down all possible observations on slips of paper, or "tickets." The observations with higher probabilities are written on many tickets. The observations with lower probabilities are written on few tickets. The tickets in the box are thoroughly mixed and one is drawn "at random" from the box. This reproducible experiment mimics, or models, the behavior of many environmental processes.
There are some simple rules for the ticket-in-a box model, or "box model," for short:
| A box has to have at least one ticket. |
| Each ticket must have a definite outcome written on it. |
| A box may be re-used by putting the ticket just drawn back in and thoroughly re-mixing all its tickets. |
An outcome, by the way, can be anything yielded by an experimental observation: a concentration, a dose, a color, a time, a whole year's worth of monitoring results, an amount of money. It depends on what you are trying to model.
When a particular outcome is written on a ticket, we will say that the ticket "has" that outcome.
1. A student wants to use a box model to describe coin flips. The experiment consists of flipping a quarter. If it does not land flat on the table, the experiment is repeated until the quarter does land flat. The face that is upward ("heads" or "tails") is recorded. What are the possible outcomes? What is written on the tickets? How many tickets are put into the box? Answer
2. A scientist wants to use a box model to describe eagle sightings. She records the numbers of eagles seen along a particular waterway each day. She believes it is impossible to see more than ten eagles on any day. What are the possible outcomes? What is written on the tickets? Answer
Each ticket in a box has an equal chance of being drawn. If there are N tickets in the box (N = 1, 2, etc.), this chance is 1/N. If exactly K of the tickets have a particular outcome written on them (K =0, 1, 2, etc.), then the chance of drawing a ticket with that outcome is K/N. Chances are often written as percentages. The percentage for K/N is 100*K/N.
Some boxes appear over and over in different applications. They--or rather the probability distributions they describe--are given standard names.
3. A box contains two tickets. One has the outcome 0, the other has the outcome 1. (The name of this box is B(1, 1/2).) What is N? How many tickets have a 0? What is the value of K? What is the chance of drawing a 0? Answer
4. A box contains four tickets. One has the outcome 0, two have the outcome 1, and one has the outcome 2. (The name of this box is B(2, 1/2).) How many outcomes are there? How many tickets are there? What is N? What is the probability of drawing a ticket with a 0? With a 1? With a 2? Answer
If we know or have guessed the probabilities we want to model, then we will need to determine how to fill the box with tickets that reproduce the desired probabilities. Suppose the outcomes are O1, O2, ..., OM and that they occur with probabilities p1, p2, ..., pM, respectively. One way to approximate these probabilities is to select a large number, N. Compute p1 * N and round that value to the nearest whole number. We will indicate the rounding process with square brackets, as in [p1 * N]. Write the outcome O1 on exactly this many tickets and put them into the box. Next, write the outcome O2 on [p2 * N] tickets and put them into the box. Repeat for the remaining outcomes.
If all the values p1 * N, p2 * N, ..., pM * N are whole numbers, then the box exactly reproduces the desired probabilities. Otherwise it only approximates them. The accuracy of the approximation increases as N gets large. We can make the box as accurate as we want by choosing a large enough quantity of tickets.
5. A meteorologist says the chance of rain tomorrow is 70 percent. What are the outcomes? What are the probabilities? Describe a box model that uses N = 10. Describe a box model using N=5. (How many tickets does it contain?) Describe a box model using N = 20. Which one is best? Answer
6. Describe a box that produces the outcome 0 with probability 1/100, the outcome 2 with probability 81/100, and the outcome 1 with probability 9/50. (This is the B(2, 0.9) box.) Answer
7. (Tricky) Describe a box that produces the outcome 0 with probability 1/2, the outcome 1 with probability 1/4, 2 with probability 1/8, and in general has the outcome K with probability 2-(K+1) for any positive integer K. (This box produces a geometric distribution with parameter 1/2) Answer
Two or more boxes can be combined. The simultaneous drawing of tickets from the boxes is the outcome of a single combined experiment.
Consider two boxes, F and G. Draw a ticket from F (read it, then replace it) and another from G. (You are not allowed to choose the box G based on what is on ticket F, nor conversely. That is, the draws must be entirely independent of each other.)
That's the experiment. Its outcome is a pair of tickets, one from box F, another from box G.
For the next several questions, let F # G be the name of this combined experiment.
8. Box F is a B(1, 1/2) box: it has two tickets, one with a 0, the other with a 1. Box G is also a B(1, 1/2) box. Because F # G is an experiment, we can describe it with a box model. What are the outcomes? How many tickets are in the box for F # G? What do they say? What are the probabilities? Answer
9. Box F contains three tickets. On them are written the words "red," "green," and "blue". Box G contains two tickets. On them are written "light" and "dark". How many tickets are in the box for G # F? What do they say on them? What are their probabilities? How is G # F related to F # G? Answer
10. Box F is a B(1, 1/3) box: it contains three tickets, one with 1, two with 0. Describe a box that models F # F. Answer
A box's contents can be transformed by renaming the outcomes. The renaming changes the outcomes but not the number of tickets in the box.
11. A box contains two tickets labeled "0" and one ticket labeled "1". What is the probability of 0? of 1? Every ticket with a "0" is now re-labeled "tails" and every ticket with a "1" is re-labeled "heads". What is in the box now? What are its outcomes? What probabilities do they have? Answer
12. A box contains four tickets. They are labeled "TT", "HT", "TH", and "HH". What is the probability of HT? Of TH? The tickets are re-labeled. The new labels specify how many times an "H" appears in the original label. What is in the box now? What are the possible outcomes? What are their probabilities? Answer
13. A box contains the usual 52 playing cards. (There are four suits of 13 cards each.) All the cards that are either hearts (one of the suits) or Jacks (one of the 13 card values) are re-labeled "1" and the rest are re-labeled "0". How many cards are now in the box? How many have 1's? How many have 0"s? What is the probability of drawing a 1? What is the probability of drawing either a heart or a Jack from a deck of cards at random? Answer
Often, tickets are relabeled after a combination of boxes is formed.
14. Two tickets are independently drawn (with replacement, as usual) from a box F containing one zero and three ones. What is the name of this box? The values on those two tickets are added. What are the possible outcomes? Describe F # F. Show how adding the values on the two tickets is a relabeling of F # F. What are the probabilities of the sums? Answer
15. A box F contains two tickets: one has a 1, another has a 2. A ticket is drawn from F with replacement. Another ticket is drawn and divided into the value of the first. For example, if a 1 is drawn and then a 2 is drawn, the result is 1/2. What are the possible outcomes? Describe the contents of a box that model the probabilities of these outcomes. Answer
The sum (or product or difference or ratio) of two boxes F and G can be defined whenever the outcomes of F and G are numbers. We form the combination F # G whose tickets contain all ordered pairs (f, g) with f drawn from F and g drawn independently from G. Then we relabel (f, g) by summing (or subtracting or multiplying or dividing) f and g. We can write this as F+G (or F-G or F*G or F/G). The only restriction is that F/G can be formed provided there is zero probability of g being 0, because you cannot divide by zero.
16. Describe a box that models the roll of a fair die. (Dice are cubes whose sides are labeled with 1, 2, ..., 6 spots.) Describe a box that models the sum showing when two dice are rolled. What are the outcomes? Are all probabilities equal? Answer
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This page was created 21 February and last updated 27 February 2001 to correct errors in the statement of problem 6.