Kamyar
In position 1, if black does not redouble, (a) black bears off 19 times out of 36 and wins 2; (b) black does not bear off 17 times out of 36 but then white misses 5 times out of 36 and black wins 2 (black doubles white out if black's previous roll was 21); (c) black does not bear off 17 times out of 36 but white does bear off 31 times out of 36 and black loses 2. On average, black wins [(19/36) x 2] + [(17/36) x (5/36) x 2] - [(17/36) x (31/36) x 2] = 0.373 points per game.
In position 1, if black does redouble, (a) black bears off 19 times out of 36 and wins 4; (b) black does not bear off 17 times out of 36 but then white doubles black out and black loses 4. On average, black wins [(19/36) x 4] - [(17/36) x 4] = 0.222 points per game.
In 1296 games, by redoubling black wins 4 instead of 2 points 684 times (19 x 36), loses 4 instead of 2 points 527 times (17 x 31) and loses 4 instead of winning 2 points 85 times (17 x 5).
Ray Kershaw