To solve the given problem, we can follow the steps below:
(a) Compute the experimental probability of rolling a 3 or 6.
First, we need to find the total number of times a 3 or a 6 was rolled. According to the results:
- Number of rolls of 3 = 93
- Number of rolls of 6 = 97
Total number of rolls of 3 or 6: \[ 93 + 97 = 190 \]
Next, we find the experimental probability. Experimental probability of rolling a 3 or 6: \[ P(3 \text{ or } 6) = \frac{\text{Number of favorable outcomes}}{\text{Total number of rolls}} = \frac{190}{500} \]
Now we can calculate: \[ P(3 \text{ or } 6) = \frac{190}{500} = 0.38 \]
So, the experimental probability of rolling a 3 or 6, rounded to the nearest thousandth, is: 0.380.
(b) Compute the theoretical probability of rolling a 3 or 6.
For a fair number cube (6-sided die), each of the six outcomes (1, 2, 3, 4, 5, 6) is equally likely. The theoretical probability for any single outcome is: \[ P(\text{single number}) = \frac{1}{6} \]
To find the theoretical probability of rolling a 3 or a 6, we can add the probabilities of each outcome: \[ P(3 \text{ or } 6) = P(3) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \] Calculating it: \[ \frac{1}{3} \approx 0.333 \]
So, the theoretical probability of rolling a 3 or 6, rounded to the nearest thousandth, is: 0.333.
(c) True statement about experimental and theoretical probability.
The statement that is true is: The larger the number of rolls, the greater the likelihood that the experimental probability will be close to the theoretical probability.
This is due to the Law of Large Numbers, which states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.