We wanted to find the probability that you beat John at your 3 sided coin game. John has 3 points and you have only one. In 2 more flips, John has a 4/9 probability of winning. In any given roll, either round or tails gives him a point. There is a 1/9 probability you will tie it up. The chances are then fifty fifty. The remaining 4/9 have John ending at 4 points. 2 of them give you 1 point, 2 give you two. When the score is 4-2, there is a 1/9 chance of getting 2 heads in a row to tie is up. With 4-3, it’s 1/3. When it’s tied up, you have a 50% chance of winning. 50% of a ninth of 2/9 is 1/81 or 2/162. 50% of a third of 2/9 is 1/27 or 6/162. And 50% of 1/9 is 1/18 or 9/162. (2+6+9) / 162 = 17/162. Not very likely at all.
Table \space of \space Probabilities\\[16pt] \begin{matrix} Points&0&1&2&3&4&5\\[8pt] 0&1\\[4pt] 1&\frac12&\frac12\\[4pt] 2&\frac16&\frac23&\frac16\\[4pt] 3&\frac1{18}&\frac49&\frac49&\frac1{18}\\[4pt] 4&\frac1{54}&\frac29&\frac{14}{27}&\frac29& \frac1{54}\\[4pt] &&&… \end{matrix}