Final answer:
Converting P(no yellows) to decimal format involves using the theoretical model with replacement, and empirical data with replacement. Theoretical probability assumes all outcomes are equally likely with each pick independent, while empirical probability relies on collected data from actual trials. Increasing the number of trials helps the empirical probability converge toward the theoretical probability.
Step-by-step explanation:
Probability Format Conversion
The difference between With Replacement and Without Replacement probabilities arises because when a sample is replaced, each pick is independent of the previous one, which maintains the original probabilities for each draw. Without replacement, each pick affects the subsequent probabilities since the composition of the sample changes. To convert the probability of picking no yellow candies (P(no yellows)) to decimal format:
- Theoretical With Replacement: To calculate this, assume all outcomes are equally likely and the sample is replaced after each pick, keeping the probability consistent.
- Empirical With Replacement: This requires actual experiment data where candies are picked and replaced, and then the ratio of occurrences without yellow candies to total picks is calculated.
The decimal values might not be close depending on the variability and size of the empirical data sample. For larger sample sizes, we'd expect empirical values to get closer to theoretical probabilities due to the
Law of Large Numbers
.
If the number of trials is increased to 240 times, the empirical probability is likely to change and become more accurate as it should converge towards the theoretical probability with a larger sample size, reflecting the expected behavior more closely.