Problem of the Summer from Professor Herschkorn Jul 24, 2015

Josh takes a coin from his pocket and decides that he will flip it 4 times in a row, writing down the outcome of each flip on a scrap of paper. After he is done flipping, he will look at the flips that immediately followed an outcome of heads, and compute the relative frequency of heads on those flips. Because the coin is fair, Josh of course expects this conditional relative frequency to be equal to the probability of flipping a heads: 0.5. Shockingly, Josh is wrong. If he were to sample 1 million fair coins and flip each coin 4 times, observing the conditional relative frequency for each coin, on average the relative frequency would be approximately 0.4.