Consider a six-sided die. Without actually rolling a die, guess the number of 1’s, 2’s, 3’s, 4’s, 5’s, and 6’s you would obtain in 50 rolls. Record your predictions in the chart below:

Predictions

\[ \begin{tabular}{|c|c|c|c|c|c|c|}\hline \# of 1's & \# of 2's & \# of 3's & \# of 4's & \# of 5's & \# of 6's & Total \\ \hline \rule [0mm]{0mm}{7mm} \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm}\\ \hline \end{tabular} \]

Now roll a die 50 times and record the number of 1’s, 2’s, 3’s, 4’s, 5’s, and 6’s you obtain.

Experimental Results

\[ \begin{tabular}{|c|c|c|c|c|c|c|}\hline \# of 1's & \# of 2's & \# of 3's & \# of 4's & \# of 5's & \# of 6's & Total \\ \hline \rule [0mm]{0mm}{7mm} \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm} & \hspace{15mm}\\ \hline \end{tabular} \] How did you come up with your predictions? How do your predictions compare with your actual results? Now make a chart to combine your data with that of the rest of the class.

Experiment 1 We investigated the results of throwing one die and recording what we saw (a , a , ..., or a ). We said that the probability of an event (for example, getting a “3” in this experiment) predicts the frequency with which we expect to see that event occur in a large number of trials. You argued the (meaning we expect to get a in about of our trials) because there were six different outcomes, only one of them is a , and you expected each outcome to occur about the same number of times.

Experiment 2 We are now investigating the results of throwing two dice and recording the sum of the faces. We are trying to analyze the probabilities associated with these sums. Let’s focus first on . We might have some different theories, such as the following:

Theory 1 .

It is proposed that a sum of was out of the possible sums .

Theory 2 .

It is proposed that a sum of was of possible results, counting as the same as :

\[ \begin{array}{cccccc} 1+1 & - & - & - & - & - \\ 2+1 & 2+2 & - & - & - & - \\ 3+1 & 3+2 & 3+3 & - & - & - \\ 4+1 & 4+2 & 4+3 & 4+4 & - & - \\ 5+1 & 5+2 & 5+3 & 5+4 & 5+5 & - \\ 6+1 & 6+2 & 6+3 & 6+4 & 6+5 & 6+6 \end{array} \]

Propose your own Theory 3.
Test all theories by computing , , …, for each theory and comparing to the dice rolls recorded by the class. What do you notice?
Which theory do you like best? Why?
How could we test our theory further?
2024-10-10 13:37:02