178.2 should be 178.4 (180.2 − 1.8) and 176.2 should be 176.6 (178.4 − 1.8)
This doesn’t change the result, though:
After 2 failed tries, even if you do have the good box, the most your net gain relative to standing pat can be is 98 additional coins.
But, the odds ratio of good box to bad box after 2 failed coins is 1:100 or less than 1% probability of good box.
So your expected gain from entering the third coin is upper bounded by (98 x 0.01) - (1 x 0.99) which is less than 0.
178.2 should be 178.4 (180.2 − 1.8) and 176.2 should be 176.6 (178.4 − 1.8)
This doesn’t change the result, though:
After 2 failed tries, even if you do have the good box, the most your net gain relative to standing pat can be is 98 additional coins.
But, the odds ratio of good box to bad box after 2 failed coins is 1:100 or less than 1% probability of good box.
So your expected gain from entering the third coin is upper bounded by (98 x 0.01) - (1 x 0.99) which is less than 0.