Failing to assign the correct probability given your information is a failure both of accuracy and of calibration.
Suppose you take a test of many multiple choice questions (say, 5 choices), and for each question I elicit from you your probability of having the right answer. Accuracy is graded by your total score on the test. Calibration is graded by your log-score on the probabilities. Our lottery enthusiast might think they’re 50% likely to have the right answer even when they are picking randomly—and because of this they will have a lower log score than someone who correctly thinks they have a 1⁄5 chance. These two people may have the same scores on the test, but they will have different scores on their ability to assign probabilities.
Failing to assign the correct probability given your information is a failure both of accuracy and of calibration.
Suppose you take a test of many multiple choice questions (say, 5 choices), and for each question I elicit from you your probability of having the right answer. Accuracy is graded by your total score on the test. Calibration is graded by your log-score on the probabilities. Our lottery enthusiast might think they’re 50% likely to have the right answer even when they are picking randomly—and because of this they will have a lower log score than someone who correctly thinks they have a 1⁄5 chance. These two people may have the same scores on the test, but they will have different scores on their ability to assign probabilities.
I have updated my post to respond to your concerns, expanding on your lottery example in particular. Let me know if I’ve adequately addressed them.