8. thinking two dice are thrown and the sum is recorded. a) if 20 trials are carried out, is it possible for…

Explanation:

Step1: Understand probability concepts

Theoretical probability is based on the nature of the event. For two - dice throws, the sample space (n(S)=6\times6 = 36) outcomes. Statistical probability is based on the results of an experiment.

Step2: Analyze part a

In 20 trials, it is not possible for statistical probabilities to equal theoretical probabilities. The theoretical probabilities for the sums of two - dice throws are fixed. For example, the sum of 2 (i.e., ((1,1))) has a theoretical probability of (\frac{1}{36}), the sum of 3 (i.e., ((1,2)) and ((2,1))) has a theoretical probability of (\frac{2}{36}), etc. Since 20 is a small number of trials, the relative frequencies (statistical probabilities) are likely to deviate from the theoretical values due to chance.

Step3: Analyze part b

The minimum number of trials for statistical and theoretical probability values to perfectly agree is a multiple of 36. This is because the sample space of throwing two dice has 36 possible outcomes. If we want the relative frequencies of all possible sums to match the theoretical probabilities exactly, the number of trials (n) must be such that when we calculate the relative frequencies (\frac{f}{n}) (where (f) is the frequency of an event), they equal the theoretical probabilities. The smallest non - zero multiple of 36 is 36 itself. So, the minimum number of trials is 36.

Answer:

a) No. Because 20 is a small number of trials and statistical probabilities are based on experimental results which are subject to chance and likely to deviate from fixed theoretical probabilities. b) 36. Since the sample space of throwing two dice has 36 possible outcomes, the minimum number of trials for exact agreement of statistical and theoretical probabilities is 36.