 The experimental probability of an event occurring can be calculated by doing many trials, and dividing the number of occurrences of the desired result by the total number of trials. For example, suppose you wanted to discover the probability of rolling a three with one die. You could roll the die 600 times, and end up with 97 threes out of the 600 rolls. This would give you an answer of 97 divided by 600, or about 0.162, or 16.2% Of course, you know that for a perfect die, the theoretical probability of rolling a three is 1 chance in 6, which is 1 divided by 6, or 16.666...% But this is only what is expected to happen. During a real experiment or roll of the die, anything can happen. The result of any single roll is not predictable. In fact, after ten rolls, or twenty, you might not get a three at all! But as the number of trials increases, the experimental result will get closer and closer the the predicted, or theoretical, answer. Our result above of 16.2% was very close to one chance in six.  Try rolling some dice yourself. We have a pair of virtual dice you can roll; look for the sum of the two dice. Try to answer these questions:
Close the window. If we are going to learn anything about which sums are more common than others, we'll have to record the sum we got after each roll. Then we can see which sums happen more often than others. We have a dice rolling simulation for you to use that keeps track of the sums you got, in a bar graph. Let's play with it first, with just one die. Leave the number of dice to roll at one, but make the number of rolls equal to five. Click on 'roll the dice' once or twice. This simulates rolling a single die five times, then five more times, as many times as you want. Do it now. Close the window. Did you notice that with only five rolls of a single die, not all the numbers came up? But the more rolls you made, the more evenly the results were spread among the six possibilities. This makes sense, since the probability that any one side of a die will come up is one in six, and the probabilities are the same for each of the six sides. Now let's try more rolls. Open the window again and try 1 die, but do 10,000 rolls at a time. Roll all 10,000 dice several times. You should notice that the number of rolls that came up on each of the six numbers gets closer and closer to being the same. (Remember that the numbers of rolls in each column are now in the thousands). Close the window.  What happens if we roll two dice?In this case, we might want to know what the sum of the two dice will be. What possibilities are there? The smallest possibility is a one on both, which gives a sum of two. The largest possibility is a six on both, which gives a sum of twelve. All sums in between two and twelve are possible. But how many different combinations are there? Here's a summary of all the possible combinations: 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 There are 36 different possible combinations of the two dice. Use the simulator again to determine which sum is most likely to occur when you roll two dice. Which sums were least likely to occur? Is there any way you could have determined the answer from the table above? Close the window.
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