Once the lead is in dummy, Declarer must draw trump and play diamonds.
The question is whether to cash hearts first. I assume that East's card
on Trick 1 is uninformative, for example that the 7 does not suggest a
singleton.
Cashing one or two hearts does not help to make the contract, and the
probability of a favourable diamond split is slightly higher than the
probability that 3 hearts can be cashed. Declarer should nevertheless
try to cash three hearts: if East ruffs then Declarer can overruff, draw
trump, and try diamonds. (If hearts are not ruffed then Declarer ruffs a
club, draws trump, and plays the top diamonds.)
To compute probabilities, I determine the relevant number of deals of
the 26 cards and subtract the number with a 6-0 trump split. For
example, the number of 5-0 diamond splits is the number we would
calculate before the opening lead minus the number of deals in which
both suits split badly with each defender holding one suit or with one
defender holding both. Let N be the number of ways to deal the
defenders' cards so each of them has at least one trump. (N =
{26choose13} - 2{6choose0}{20choose13} = 10,245,560.) The probability of
a 5-0 diamond split is
{2over N} ( {5choose0}{21choose13} - {5choose0}{6choose0}{15choose13} -
{5choose0}{6choose6}{15choose7} )
= {393,900 over N} approx .0384459 .
The probability of success by running diamonds is approximately .9615541
.
Note that a defender cannot have 8 hearts and 6 trumps. The probability
of an 8-2 or worse heart split is therefore
{2over N} ( {10choose0}{16choose13} + {10choose1}{16choose12} +
{10choose2}{16choose11} - {10choose0}{6choose6}{10choose7} -
{10choose1}{6choose6}{10choose6} - {10choose2}{6choose6}{10choose5} )
= {403,520 over N} approx .0393849 .
The probability that the hearts can be cashed is approximately .9606151
. The excellent possibility of recovering if East ruffs makes this the
correct play, but puzzle lovers may want more detail.
The number of deals in which East has two hearts is
{10choose2}{16choose11}. We must subtract the number of those in which
trumps or diamonds split badly. The inclusion-exclusion calculation is
simplified by two observations: a defender cannot have 2 hearts and 0
trumps, and a defender with 2 hearts and 0 diamonds must have all the
outstanding black cards. The probability that Declarer can make despite
a third-round heart ruff is therefore
{{10choose2} over N} ( {16choose11} - {5choose5}{11choose6} -
{6choose6}{10choose5} + {5choose5}{6choose6}{5choose0} )
= {164,475 over N} approx .0160533 .
Similar calculations for the 9-1 & 10-0 heart splits show that the
probability of making by trying to cash hearts is
{10,019,650 over N} approx .9779504 .