We toss a fair coin 10000 times and record the sequence of the results. Then we count the number of times that a sequence of 5 heads in a row followed immediately by 5 tails in a row has occurred among these results. (Of course, this number is a random variable.) What is the expected value of this number?
Enter your answer as a decimal or a fraction, whichever you prefer.
I have made some progress on it but granted I have one attempt left I didn't want to mess this up. I have determined that for 5 heads, and for 5 tails to occur in ten tries, the probability is 0.0009765625 with an expected value as well. My line of thinking was since we can't expect to get this sequence occur until the 10th try, the expected value of flipping 10,000 times would be 9990*0.0009765625 but this was wrong. I feel I'm very close since for any sequence of ten tries, the expected value will be 0.0009765625
Best Answer
$$N=10000,\ n=10\implies\frac{N-n+1}{2^n}=\frac{9991}{1024}\equiv9.7568359375$$