There is a number cube with faces numbered 1 to 6 . There is also a coin with one side marked as heads and the other tails. As a trial of an experiment, the number cube was rolled and the coin flipped. The number ( 1 to 6$)$ from the roll and the side ( $H$ for heads and $T$ for tails) of the coin from the flip were recorded. Here is a summary of the data from 825 trials. \begin{tabular}{|r|c|c|c|c|c|c|c|c|c|c|c|c|} \hline Outcome & $1 H$ & $2 H$ & $3 H$ & $4 H$ & $5 H$ & $6 H$ & $1 T$ & $2 T$ & $3 T$ & $4 T$ & $5 T$ & $6 T$ \\ \hline Number of trials & 69 & 71 & 66 & 67 & 73 & 69 & 71 & 69 & 65 & 63 & 68 & 74 \\ \hline \end{tabular} Answer each part. (a) Assuming the cube and the coin are fair, find the theoretical probability of this event: both rolling a 1 or a 4 and flipping tails, in a single trial. Round your answer to the nearest thousandth. (b) Use the data to find the experimental probability of this event: both rolling a 1 or a 4 and flipping tails, in a single trial. Round your answer to the nearest thousandth.
Solution
Step 1 ：Given that the cube and the coin are fair, each outcome of the cube (1 to 6) and the coin (H or T) are equally likely. The event we are interested in is rolling a 1 or a 4 and flipping tails. This means we have two favorable outcomes (1T and 4T) out of a total of 12 possible outcomes (6 numbers times 2 sides of the coin).
Step 2 ：Calculate the theoretical probability by dividing the number of favorable outcomes by the total number of outcomes. Theoretical probability = \(\frac{2}{12} = 0.16666666666666666\)
Step 3 ：Round the theoretical probability to the nearest thousandth to get approximately 0.167.
Step 4 ：Use the data from the trials to calculate the experimental probability. The favorable outcomes are the number of trials that resulted in 1T or 4T. The total number of trials is the sum of all trials.
Step 5 ：Calculate the experimental probability by dividing the number of favorable outcomes by the total number of outcomes. Experimental probability = \(\frac{134}{825} = 0.16242424242424242\)
Step 6 ：Round the experimental probability to the nearest thousandth to get approximately 0.162.
Step 7 ：Final Answer: The theoretical probability of rolling a 1 or a 4 and flipping tails, in a single trial, is approximately \(\boxed{0.167}\). The experimental probability of the same event, based on the data from the trials, is approximately \(\boxed{0.162}\).
