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Please watch video below. What is the theoretical probability of rolling a fair die and obtaining a 1? If you roll a fair die 6 times, how many times would you expect, theoretically, of obtaining a 1? What if you performed an experiment and rolled the die 6 times. Will you definitely obtain your theoretical answer?

Why is experimental probability often different than theoretical probability? What do you think you should expect if you increase the number of trials, that is, rolls of the die? Why?

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The theoretical probability of rolling a 1 on a fair six-sided die is calculated by considering the number of favorable outcomes over the total number of possible outcomes. Since the die is fair and has six sides numbered from 1 to 6, the probability of rolling a 1 in a single roll is 1/6. This probability assumes all outcomes are equally likely and that the die is unbiased.

When rolling the die six times, the expected number of times a 1 will appear, based on theoretical probability, can be determined by multiplying the probability of obtaining a 1 in a single roll by the total number of rolls. Therefore, the expected number of 1s in 6 rolls is (1/6) × 6 = 1. This means that, on average, over many sets of six rolls, a 1 will appear once per set, according to the theoretical probability.

However, if you perform an actual experiment by rolling the die six times, the number of 1s you observe may not exactly match the expected value of 1. This discrepancy arises because of the variability inherent in random experiments, known as variability or sampling error. Each series of six rolls is a small sample size, and individual outcomes may deviate from what theory predicts merely due to chance. For instance, you might roll no 1s, one 1, or even two 1s in a particular trial, even though the expected average is one. Over a larger number of trials, these fluctuations tend to even out, and the experimental probability tends to approach the theoretical probability.

Experimental probability is often different from theoretical probability because real-world outcomes are influenced by randomness, chance, and small sample sizes. While the theoretical probability assumes an ideal scenario with an infinite number of trials, actual experiments are limited and can be affected by bias or imperfect randomness. As the number of trials increases, the law of large numbers states that the experimental probability will tend to converge to the theoretical probability. In other words, the more times you roll the die, the closer your observed proportion of 1s will be to the expected 1/6.

In conclusion, understanding the relationship between theoretical and experimental probabilities provides insight into how randomness behaves in practice. While small samples can exhibit significant fluctuations, increasing the number of trials improves the accuracy of the experimental probability, making it a reliable approximation of the theoretical value. This principle underpins many statistical methods and confirms the importance of large sample sizes in obtaining dependable results.

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