EXAMPLE 4 What looks random?

Toss a fair coin six times and record heads (H) or tails (T) on each toss. Which of these outcomes is most probable?

HTHTTH  HHHTTT  TTTTTT

Almost everyone says that HTHTTH is more probable because TTTTTT and HHHTTT do not “look random.” In fact, all three are equally probable. That heads and tails are equally probable says all specific outcomes of heads and tails in six tosses are equally likely. That heads and tails are equally probable says only that about half of a very long sequence of tosses will be heads. This is because in very long sequences of tosses, the number of outcomes for which the proportion of heads is approximately one-half is much larger that the number of outcomes for which the proportion is not near one-half. That heads and tails are equally probable doesn’t say that heads and tails must come close to alternating in the short run. It doesn’t say that specific outcomes that balance the number of heads and tails are more likely than specific outcomes that don’t. The coin has no memory. It doesn’t know what past outcomes were, and it can’t try to create a balanced sequence.