Exercises

1. A standard six-sided die is rolled during two different experiments. Determine the theoretical probability of rolling a multiple of \(3\) and calculate the two estimates of this probability that arise from the experimental data shown. Compare the three values that you calculated.

   	Experiment 1						Experiment 2
   Outcome (Roll)	1	2	3	4	5	6	1	2	3	4	5	6
   Observed Frequency	\(8\)	\(10\)	\(5\)	\(15\)	\(9\)	\(13\)	\(12\)	\(6\)	\(10\)	\(5\)	\(3\)	\(4\)

2. Explain how buying at least one lottery ticket every time a lottery is held might be used to estimate the probability of winning the lottery. Do you think this experimental method is a good strategy for estimating this probability? Identify some advantages and disadvantages of this method.

3. A certain device will randomly produce either a \(0\) or a \(1\). The probability that the device will produce a \(1\) on any given trial is unknown. To estimate the value of this probability, experiments were run using the device. Each experiment consisted of \(50\) trials and a total of \(20\) experiments were run. The results of the \(20\) experiments are shown in the table below.

   Experiment Number	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
   Number of 1s Observed	\(20\)	\(17\)	\(14\)	\(19\)	\(17\)	\(18\)	\(16\)	\(15\)	\(19\)	\(22\)	\(16\)	\(17\)	\(19\)	\(19\)	\(18\)	\(20\)	\(18\)	\(19\)	\(25\)	\(20\)

   1. Calculate the probability estimate for each of the first \(10\) experiments. Display these \(10\) pieces of data in a graph.
   2. How does the graph in part a) change if instead of plotting each experiment individually, you plot the cumulative data after each new experiment? For example, after the first experiment we have performed \(50\) trials, \(20\) of which were favourable, and after the second experiment we have performed \(100\) trials, \(37\) of which were favourable, and so on.
   3. Use the given data to determine a single estimate for the probability that the device will produce a \(1\).
4. Jumana wanted to estimate the probability that a particular paper cup will land open end down when tossed. After running an experiment with exactly \(50\) trials, Jumana calculated an experimental probability of \(29\%\) and used this as an estimate for the true probability. Explain why Jumana must have made a mathematical mistake during the experimental probability calculation.
5. Jun experimented by tossing a coin. The results of each experiment are shown below.

   Number of Trials	\(20\)	\(50\)	\(50\)	\(25\)	\(100\)	\(50\)	\(60\)	\(25\)	\(50\)	\(100\)
   Number of Heads	\(12\)	\(30\)	\(26\)	\(12\)	\(47\)	\(18\)	\(39\)	\(5\)	\(34\)	\(56\)

   Calculate all \(10\) experimental probabilities of tossing heads and determine the mean, median, and mode of the \(10\) values. Which of the three measures is closest to the theoretical probability of tossing heads on a fair coin?
6. You have two spinners. One spinner has three sections labelled \(1,2,3\) and the other spinner has twenty sections labelled with the integers \(1\) through \(20\). You would like to estimate the theoretical probability of landing on the number \(1\) on the two different spinners. Since the spinner with twenty sections has many more outcomes than the spinner with three sections, it might seem as though more trials ought to be necessary to accurately estimate the probability of interest for the spinner with twenty sections. Do you think this is true? Explain your reasoning.
7. You have one spinner with three sections labelled \(A,B,C\). The section labelled \(A\) appears to be around half the total area of the spinner and the section labelled \(B\) is very small in size. You would like to estimate the theoretical probability of landing on \(A\) and the theoretical probability of landing on \(B\). If you were to perform an experiment to estimate the values of these probabilities, then do you think you would need more trials to get accurate information about landing on \(A\) or about landing on \(B\)? Explain your reasoning.