A chance process that generates a number can be modeled by drawing tickets
from a box.
To model a chance process with a box model, we find an analogy between the process and drawing
numbers at random from a box.
Rolling a die is like drawing from a box containing tickets labeled with the numbers 1, 2, 3, 4, 5, and 6.
If 1 represents heads and 0 tails, flipping a coin is like drawing a ticket from a box containing the numbers 0 and 1.
Repetitions of the chance process can be modeled with repeated draws from the box. If the same outcomes are possible with each draw (as
in the case of a die roll or coin toss) draws are made with replacement.
A box model is an analogy between a chance process and drawing tickets from a box.
With replacement indicates that after a ticket is drawn it is replaced in the box before another draw is made.
To make a box model:
Decide which numbers go into the box.
Decide how many of each of number go into the box.
Decide how many draws are appropriate.
Decide whether the draws are made with or without replacement.
Example: Create a box model to simulate rolling a six-sided die 4 times.
Decide which numbers go into the box.
When rolling a six-sided die, the possible outcomes are 1, 2, 3, 4, 5, and 6, so these
are the numbers that go into the box.
Decide how many of each kind of number go into the box.
Each of the numbers occurs once on the die so there should be one of each in the box.
Decide how many draws are appropriate.
We're rolling the die 4 times, so there will be 4 draws.
Decide whether the draws are made with or without replacement.
Each time we roll the die, the same outcomes are possible, thus we draw with
replacement.
Use the applet to create a box model.
Click on the buttons in the number pad to add tickets
to the box.
Choose the number of draws that will be made.
All draws will be made with
replacement.
Example: Create a box model to simulate rolling a six-sided die 5 times and counting the
number of 3's.
Decide which numbers go into the box.
If we roll a 3, the number of 3's increases by 1 otherwise it does not increase, thus
the numbers in the box are 1 and 0.
Decide how many of each kind of number go into the box.
There is only 1 threes but 5 non-threes so the box would contain 1 1 and 5 0's.
Decide how many draws are appropriate.
We're rolling the die 5 times, so there will be 5 draws.
Decide whether the draws are made with or without replacement.
Each time we roll the die, the same outcomes are possible, thus we draw with
replacement.
Decide which numbers go into the box.
Solution: The numbers are 0 and 1. Rolling a head is like drawing a 1 from the box.
Decide how many of each kind of number go into the box.
Solution: There is one head and one tail on the coin thus there is a 1 and a 0 in the box.
Decide how many draws are appropriate.
Solution: Flipping the coin twice is like to 2 draws from the box.
Decide whether the draws are made with or without replacement.
The same outcomes are possible with each coin flip thus the draws are made with
replacement.