Objectives

1. Predict the number of 2’s and 3’s that will come up if 10 six-sided dice are rolled and compare to the results of repeated experiments.
2. Predict the most probable outcome when rolling two six-sided dice and adding them together, and compare to repeated experiments.

Resources

• ~20 six-sided dice

Background

When performing an experiment with a given number of possible outcomes, all of which are equally likely, the probability of a given outcome is simply the number of ways for that outcome to occur divided by the total number of possible outcomes. When rolling a single 6-sided die this is simple, as the likelihood of any result is just 1/6=16.7%. However, in the case of adding two die this becomes more complicated, as there are multiple ways to achieve some outcomes. For example – 4 can be the result of rolling a 1 and a 3, or two 2’s.

If an experiment that has an outcome X with a probability p is performed N times, the the number of times X occurs will on average be Np. However, because this is probability, the actual number won’t always be Np – sometimes it will be more, sometimes less. The standard deviation of the number of times X would occur is given by \sigma_X=\sqrt{Np(1-p)}

Method

1. Roll ten dice and record the number of 2’s and 3’s. Repeat this at least 20 times. Count the number of times you get each result (the frequency), and plot the frequency vs. the result. Compute the average and standard deviation of your data and compare to the predicted values.
2. Roll twenty dice, add pairs together, and record the number of 7’s. Repeat this at least 20 times. Plot the frequency vs. the outcome, compute the average and standard deviation, and compare to predicted values.

Analysis Questions

1. How probable is your outcome for each experiment above, based on the number of standard deviations from the expected value?
2. How is this probability different from the probability of getting the expected (i.e. average) value Np? Why is the probability of getting the expected value not 100%?