Part A: Theoretical Probability of a Fair Coin Landing on Heads

The theoretical probability of a fair coin landing on heads is calculated by considering the total number of possible outcomes and the number of favorable outcomes.

• A fair coin has two possible outcomes: heads or tails.
• There is only one favorable outcome for heads.

The theoretical probability (P) is given by: $P(\text{heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{2} = 0.5$

So, the theoretical probability of a fair coin landing on heads is 0.5, or 50%.

Part B: Experimental Probability of Landing on Heads

To determine the experimental probability, we need to perform the experiment of flipping a coin 12 times and recording the frequency of each outcome. Let's assume the coin flips yield the following sequence:

H, T, H, H, T, H, T, T, H, T, H, H

Frequency of outcomes:

• Heads (H): 7 times
• Tails (T): 5 times

The experimental probability (P) of landing on heads is given by: $P(\text{heads}) = \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{7}{12} \approx 0.583$

So, the experimental probability of landing on heads is approximately 0.583, or 58.3%.

Part C: Comparison of Experimental Probability to Theoretical Probability

The theoretical probability of landing on heads is 0.5 (50%), whereas the experimental probability from our 12 coin flips is approximately 0.583 (58.3%).

Comparison:

• The theoretical probability (0.5) represents the expected probability based on a fair coin.
• The experimental probability (0.583) is based on actual outcomes from 12 flips.

In this case, the experimental probability is close to but slightly higher than the theoretical probability. This difference can be attributed to the relatively small number of trials (12 flips). As more flips are conducted, the experimental probability is expected to get closer to the theoretical probability due to the Law of Large Numbers.

Would you like more details on any part of this topic, or do you have any questions?

Here are 8 related questions for further exploration:

1. How does the Law of Large Numbers explain the convergence of experimental probability to theoretical probability?
2. What factors can cause discrepancies between experimental and theoretical probabilities?
3. How can you increase the accuracy of experimental probability in an experiment?
4. What is the difference between relative frequency and probability?
5. Can experimental probability ever be exactly equal to theoretical probability?
6. What are some examples of real-life situations where theoretical and experimental probabilities are used?
7. How does sample size impact the reliability of experimental results?
8. What steps can be taken to ensure fairness in probability experiments?

Tip: When conducting probability experiments, it's important to ensure that each trial is independent and that the conditions remain constant to achieve accurate results.