A six-sided die of unknown bias is rolled 20 times, and the number 3 appears six times. In the next three sets of 20 rolls, the number 3 comes up six times, five times, and seven times, respectively. The experimental probability of rolling a 3 is __%, which is approximately __% higher than its theoretical probability. (Round your answers to the nearest whole number.)

To find the experimental probability of rolling a 3, you would take the number of times a 3 appears (which is 6 + 5 + 7 + 6 = 24 times) and divide it by the total number of rolls (which is 20 rolls x 4 sets = 80 rolls).

Experimental probability = Number of times a 3 appears / Total number of rolls Experimental probability = 24 / 80 Experimental probability = 0.3

To convert this probability into a percentage, you multiply by 100. 0.3 x 100 = 30%

So, the experimental probability of rolling a 3 is 30%.

For a standard six-sided die, the theoretical probability of rolling a 3 (or any other single number) is 1/6, since there is one 3 on a die and six possible outcomes.

Theoretical probability = 1 / 6 Theoretical probability ≈ 0.1667

Converting to a percentage: 0.1667 x 100 ≈ 16.67%

Now, let's calculate how much higher the experimental probability is compared to the theoretical probability.

Experimental probability - Theoretical probability = 30% - 16.67% = 13.33%

Rounding to the nearest whole number, the experimental probability is approximately 13% higher than the theoretical probability.

Extra: The theoretical probability is based on the assumption that the die is fair and unbiased, that every outcome has an equal chance of occurring on each roll. This is a fundamental concept in probability theory known as equally likely outcomes. If the die is perfectly balanced and unbiased, each of the six faces should appear about one-sixth of the time over a large number of rolls.

On the other hand, the experimental probability is calculated from the actual results of rolling the die. It is specific to the particular die and the specific sequence of rolls performed in the experiment. If the die is biased, the experimental probability may not match the theoretical probability. Fluctuations can also happen simply due to chance, especially when the number of trials (in this case, rolls of the die) is not large enough to even out the variations.

In educational settings, understanding the difference between experimental and theoretical probability helps students grasp the concept of variation in experiments and the importance of a large sample size. With a larger number of trials, the experimental probability should get closer to the theoretical probability if the die is not biased. This demonstrates the law of large numbers, a principle that underlies many statistical concepts.