Final answer:
The probability of getting heads when flipping a fair coin is 0.5, and for consecutive heads in multiple flips, multiply 0.5 for each flip (e.g., for two flips, it’s 0.5 x 0.5 = 0.25). In the long term, the relative frequency of getting heads should approach the theoretical probability due to the law of large numbers, as demonstrated by Karl Pearson's coin-toss experiment.
Step-by-step explanation:
When you flip a fair coin, the probability of landing on heads is always 0.5, or 50%. This is true for every individual flip no matter how many times you flip the coin since each flip is an independent event. When considering multiple flips in a row, the chance of getting heads on each flip still remains 0.5. However, if you're interested in the probability of obtaining heads in a row, for instance, twice in a row, you would calculate this by multiplying the probability of getting heads on each individual flip. So, for two flips, this would be 0.5 x 0.5, which equals 0.25 or 25%. For three consecutive heads, it would be 0.5 x 0.5 x 0.5, which equals 0.125 or 12.5%, and so on.
The implication of this theoretical probability is that the results for a small number of tosses can vary greatly, but as you increase the number of coin tosses, the relative frequency of the outcomes will approach 50% heads and 50% tails, which is supported by the law of large numbers. Karl Pearson's experiment flipping a fair coin 24,000 times obtained 12,012 heads, illustrating that the relative frequency, 0.5005, was very close to the theoretical probability of 0.5.