Your intuition's not bad. The expected value for the longest run of heads in N total flips of a fair coin is around log2(N) - 1 with a standard deviation that's

Your intuition's not bad. The expected value for the longest run of heads in N total flips of a fair coin is around log2(N) - 1 with a standard deviation that's approximately 1.873 plus a term that vanishes as N grows large. log2(10B) - 1 is approximately 32 and with that standard deviation, even a run of 100 in 10B flips is incredibly unlikely. For more info see Mark F. Schilling's paper, "The Longest Run of Heads" available here https://www.csun.edu/~hcmth031/tlroh.pdf https://www.csun.edu/~hcmth031/tlroh.pdf.