The Gambler's Fallacy and the Hot-Hand Fallacy

A fair coin is flipped 100 times. (a) Conditional on the coin having just landed Heads three times in a row, what is the probability the next flip is Heads? A gambler insists it should be less than $1/2$ because the coin 'is due for Tails.' Explain the error precisely. (b) Now consider a different setting: a basketball player's shots are modeled as a Markov chain where $P(\text{Hit} \mid \text{previous Hit}) = 0.6$ and $P(\text{Hit} \mid \text{previous Miss}) = 0.4$. After three consecutive hits, what is the probability the next shot is a hit? (c) Miller and Sanjurjo (2018) showed that even with a fair coin, if you select flips that follow a streak of $k$ Heads and compute the sample proportion of Heads among those selected flips, the expected proportion is strictly less than $1/2$. Why does this happen, and what does it imply for empirical hot-hand studies?