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Click Setup to design an experiment where a block is drawn from a bag containing 2 or more (up to 10,000) colored blocks. There are two versions of the simulation: Basic and DrawUntil.
In both versions of Blocks in a Bag:
In this simulation, 2 target points are chosen on a number line – one to the left of a starting point (0) and one to the right. You begin by taking one step, left or right, from the starting point, then continue taking steps from there. The chances of moving left or right for each step taken are given in the setup. The question you explore is: “How many steps does it take to reach one of the target points?”
Click Setup to drag the left (negative) and right (positive) target points from the starting point (0). Do this by dragging the points on the number line. The maximum number of steps in either direction is 10, but the targets don’t have to be the same distance from 0.
Then set the chances of moving left or right for each step taken by picking one of 3 options:
When your setup is complete click Walk Once to simulate one random walk or Walk from 2 to 10,000 times. Check “Animation” to see the point step along the number line. Click “Show chance of moves” to see the set chances of moving right or left for each step taken.
Whether your outcome is to walk once or walk several times, the results table and graph for previous walks are cleared each time you click one of the buttons. Check “Accumulate Results” to add new walks to the ones you have already done; in this way you can display results of your randomwalk setup indefinitely. Note that within the frequency table and graph windows you have options for other graphs and display formatting like those described earlier in this Help document.
The Estimate Area simulation lets you use randomness and the power of a computer to estimate areas of figures drawn on screen instead of counting squares or using area formulas. The idea is to drop many tiny dots on the drawing and tally the colors they land on. The percent of dots that land on each color is an estimate of the percent of the whole drawing area that has that color. To help you understand the idea, the whole drawing area includes 10 × 10 = 100 smaller squares, so each smaller square you color is 1 percent of the total. For example, in the drawing at right, there are 2 small gray squares so they make up 2 percent of the whole drawing area.
To run a simulation, click on any of the 5 colors to set your “paintbrush” and draw. You can draw over any color with any other color to fix mistakes or edit your drawing. Click Show Grid if that helps you place parts of your drawing. When you are finished drawing, click Drop Once to drop the first random blue dot. (Look carefully, they are tiny.) Of course, the idea is to drop many, many dots – the more you drop the better the area estimate. So you can also set it to Drop 2 to 10,000 dots at a time. Click Clear to remove the dropped dots from your drawing. Click Erase Paint to clear your drawing.
The next illustration shows the results of dropping 500 dots on the drawing. Note that in this run, the estimate of gray in the drawing is 1 percent. The estimate of brown in the drawing is 13 percent. Count small colored squares or click Show Color Percents to see the actual percents of colors in the drawing.
Whether your outcome is to drop once or drop several times, the results table and graph are cleared each time you click one of the buttons. Check “Accumulate Results” to drop new dots on top of the ones you already have; in this way you can display results of your area estimation setup indefinitely. Note that within the frequency table and graph windows you have options for other graphs and display formatting like those described earlier in this Help document.
Manfred Eigen is a chemist and physicist who studies, among other things, how populations grow while individuals live and die. An indifferent strategy model for growth is one in which the birth or death of an individual is independent of the current state of the population. (Compare this to the equilibrium growth model described in another simulation.) The Indifferent Strategy simulation lets you examine the longterm results of this model.
You start with a simplified population represented by 2sided blue/yellow markers filling a square game board. Click Setup to define the population by choosing the size of the game board (population) – from 2 × 2 to 10 × 10 squares. You can then decide how many blue and how many yellow markers are in the starting population. Note that the total number of markers must be equal to the number of squares on the board. The default setup is a 4 × 4 board with 8 blue and 8 yellow randomly placed markers.
When your setup is complete click Flip Once to flip a single marker. The simulation starts by flipping a coin. If the result is a Head, a randomly chosen blue marker is flipped to blue. If the result is a Tail, a randomly chosen yellow marker is flipped to blue. The flip is ignored if there are no markers that match the flip. You might think of flipping from yellow to blue as a birth and flipping from blue to yellow as a death – so a Head means a death, and a Tail means a birth.
You might find it helpful to think of 1 flip as 1 generation of the population. Whether you Flip Once or Flip 2 to 10,000 times you will see the tally of different numbers of blue markers on the board over that number of generations. You may need to scroll down the frequency table to see some results.
The question is, “Under the indifferent strategy model for any given starting population is there a predictable number of live members of the population in the long run?” In terms of blue/yellow markers, “For any number of blue markers in the starting board, is there a predictable number of blue markers after many flips?” “Does it matter where the markers are on the board?” What do you think?
Whether your outcome is to flip once or flip several times, the results table and graph are cleared each time you click one of the buttons, but the next flip or flips will begin with the current board. Click Clear to start over with a random new board with your setup; this also clears the results table and graph. Check “Accumulate Results” to add new flips to the ones you have already done; in this way you can display results of your indifferent strategy setup indefinitely. Note that within the frequency table and graph windows you have options for other graphs and display formatting like those described earlier in this Help document.
Manfred Eigen is a chemist and physicist who studies, among other things, how populations grow while individuals live and die. An equilibrium model for growth is one in which the birth or death of an individual depends on the current state of the population. (Compare this to the indifferent strategy model described in another simulation.) The Equilibrium simulation lets you examine the longterm results of this model.
You start with a simplified population represented by 2sided blue/yellow markers filling a square game board. Click Setup to define the population by choosing the size of the game board (population) – from 2 × 2 to 10 × 10 squares. You can then decide how many blue and how many yellow markers are in the starting population. Note that the total number of markers must be equal to the number of squares on the board. The default setup is a 4 × 4 board with 8 blue and 8 yellow randomly placed markers.
When your setup is complete click Flip Once to flip a single random marker. In this simulation, a specific marker (member of the population) is picked at random and then flipped – if it is alive (blue) it dies (flips to yellow), and if it is not alive (yellow) it is born (flips to blue). (The computer picks a random row and a random column in the game board to pick the specific marker.)
You might find it helpful to think of 1 flip as 1 generation of the population. Whether you Flip Once or Flip 2 to 10,000 times you will see the tally of different numbers of blue markers on the board over that number of generations. You may need to scroll the frequency table and graph to see more data. The question is, “Under an equilibrium model, for any given starting population is there a predictable number of live members of the population in the long run?” In terms of blue/yellow markers, “For any number of blue markers on the starting board, is there a predictable number of blue markers after many flips?” “Does it matter where the markers are on the board?” What do you think?
Whether your outcome is to flip once or flip several times, the results table and graph are cleared each time you click one of the buttons, but the next flip or flips will begin with the current board. Click Clear to start over with a random new board with your setup; this also clears the results table and graph. Check “Accumulate Results” to add new flips to the ones you have already done; in this way you can display results of your indifferent strategy setup indefinitely. Note that within the frequency table and graph windows you have options for other graphs and display formatting like those described earlier in this Help document.