Questions: Assuming that the cube is fair, choose the statement below that is true. The larger the number of rolls, the greater the likelihood that the experi will be close to the theoretical probability. The experimental probability will never be very close to the theoretical p the number of rolls. The smaller the number of rolls, the greater the likelihood that the exper will be close to the theoretical probability.

Solution Steps

To determine which statement is true, we need to understand the relationship between the number of rolls of a fair cube and the experimental probability approaching the theoretical probability. The Law of Large Numbers states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.

We need to determine which statement about the relationship between the number of rolls of a fair cube and the experimental probability is true. The Law of Large Numbers suggests that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.

For a fair six-sided die, the theoretical probability of each face (1 through 6) is: \[ P(\text{face}) = \frac{1}{6} \approx 0.1667 \] Thus, the theoretical probability distribution is: \[ [0.1667, 0.1667, 0.1667, 0.1667, 0.1667, 0.1667] \]

Given the experimental probabilities after 1000 rolls: \[ [0.15, 0.169, 0.169, 0.167, 0.19, 0.155] \]

We compare the experimental probabilities with the theoretical probabilities: \[ \begin{align_} 0.15 & \quad \text{vs} \quad 0.1667 \\ 0.169 & \quad \text{vs} \quad 0.1667 \\ 0.169 & \quad \text{vs} \quad 0.1667 \\ 0.167 & \quad \text{vs} \quad 0.1667 \\ 0.19 & \quad \text{vs} \quad 0.1667 \\ 0.155 & \quad \text{vs} \quad 0.1667 \\ \end{align_} \]

The experimental probabilities are close to the theoretical probabilities, and as the number of rolls increases, the experimental probabilities are expected to get even closer to the theoretical probabilities. This supports the statement: "The larger the number of rolls, the greater the likelihood that the experimental probability will be close to the theoretical probability."

Final Answer