Final answer:
The correlation between the number of heads in the first 100 coin tosses (H100) and the total number of heads after 300 coin tosses (H300) is approximately 0.577, indicating a positive linear relationship.
Step-by-step explanation:
The correlation between H100 and H300 represents the relationship between the number of heads in the first 100 tosses of a coin and the total number of heads after 300 tosses. To calculate this, we must consider that H300 is composed of H100 and the number of heads in the remaining 200 tosses, H200. Since H100 and H200 are independent, the correlation coefficient formula can be simplified as Corr(H100, H300) = Var(H100) / sqrt(Var(H100) * Var(H300)).
For a fair coin, the probability of getting a head in any toss is 0.5. Therefore, the expected number of heads in H100 is 50, and in H300 is 150. The variance of a binomial distribution with probability p and n trials is np(1-p). Therefore, Var(H100) = 100*0.5*0.5 = 25 and Var(H300) = 300*0.5*0.5 = 75.
Putting these into the correlation coefficient formula, we get Corr(H100, H300) = 25 / sqrt(25*75) = 25 / (5*sqrt(75)) = 25 / (5*5*sqrt(3)) = 1/sqrt(3) ≈ 0.577. Thus, the correlation between H100 and H300 is approximately 0.577, indicating a positive linear relationship between the number of heads in the first 100 tosses and the total number of heads after 300 tosses.