Box Model
Click on the number pad to place numbers in the box. Press Start ; numbers from your box will be randomly drawn with replacement.
Each time a number is drawn, the frequency chart is updated. Select "Show Theoretical Probability." As you continue to draw from the box, the relative frequencies get closer and closer to the actual probabilities, a visualization of the Law of Large Numbers.
You can use this manipulative to explore many different chance processes. For example, consider tossing a fair coin. Place a 0 and a 1 in the box and let heads be represented by the number 1 and tails by the number 2. Click on Start and observe the results of repeatedly tossing a fair coin.
To toss a biased coin which comes up heads twice as often as it does tails, place two 1's and one 0 in the box.
Rolling a pair of fair dice and observing the sum is another chance process that can be simulated by using Box Model. There are thirty-six equally likely outcomes when rolling a pair of dice:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6).
Note that three of the thirty-six outcomes, (1, 3), (2, 2), (3, 1), yield a sum of 4, so we want three 4's in the box. All told, we place one 2, two 3's, three 4's, four 5's, five 6's, six 7's, five 8's, four 9's, three 10's, two 11's, and one 12 in the box. Press Start and you are rolling the dice!