b. Would an empirical probability based on 100 games or 1000 games give a better estimate of the probability of winning Solitaire? Why?

Verified step by step guidance

1. Understand that empirical probability is calculated as the ratio of the number of successful outcomes to the total number of trials, i.e., \(\text{Empirical Probability} = \frac{\text{Number of Wins}}{\text{Total Games Played}}\).
2. Recognize that the Law of Large Numbers states that as the number of trials increases, the empirical probability tends to get closer to the true theoretical probability.
3. Compare the two scenarios: 100 games versus 1000 games. A larger number of games (1000) provides more data points and reduces the effect of random fluctuations or anomalies.
4. Conclude that the empirical probability based on 1000 games would generally give a better estimate of the true probability of winning Solitaire because it is more stable and reliable due to the larger sample size.
5. Remember that while increasing the number of games improves the estimate, practical constraints such as time and resources should also be considered.

Key Concepts

Empirical Probability

Empirical probability is the likelihood of an event based on observed data from experiments or trials. It is calculated by dividing the number of times the event occurs by the total number of trials. This approach relies on actual outcomes rather than theoretical assumptions.

Law of Large Numbers

The Law of Large Numbers states that as the number of trials increases, the empirical probability tends to get closer to the true theoretical probability. This means that larger sample sizes generally provide more accurate and stable estimates of probability.

Sample Size and Estimation Accuracy

Sample size affects the reliability of probability estimates; larger samples reduce random fluctuations and sampling error. Therefore, an empirical probability based on 1000 games is typically a better estimate than one based on 100 games because it is less influenced by chance variations.