This is true, but not what it is about. What we try to establish is the expected value of the proportion of heads after a preceding heads in a sequence of three flips.
The proportions in the table are, respectively, 0, 1, 0, 0, 0.5 and 1. Each of these proportions has a likelihood of 1/6 to occur (since the six sequences concerned are equally likely). The expected value is therefore the sum of these six proportions divided by six, which gives 5/12.
You can simulate this easily in excel — I just did it with 20 sequences of 20 flips:
The leftmost part is the outcomes. The middle bit is what Jack writes down (i.e. the outcomes after flipping heads). The next two columns show the number of heads and the number of tails, the last columns shows the proportion of heads of the total. The bold number at the top is the average of the twenty flips.
I ran this 30 times, 22 times out of which the average proportion over 20 flips was below 0.5, 8 times it was above.
Hope this helps!