This puzzle is a modification of a problem from the William Lowell Putnam Mathematics Competition.
A particular magic coin has the following strange property: on any given day, the probability that the coin lands "heads up" is exactly the proportion of heads tossed that day (that is, after the coin landed heads up and tails up at least once). So for example, if you have tossed the coin 10 times today, and it has landed heads up 4 times, the probability that the coin will land heads up on the next toss is 0.4.
One day you toss the coin twice, and it lands heads up and tails up once each. If you toss the coin 98 more times that day, which number of heads is most likely to appear?