Mutual information does not change based on which of the two random values is revealed. This means we know just as much about "temperature when told the month" as we know about "the month when told the temperature".
m j uː t̠ʃ uː ə l WORD_BOUNDARY ɪ n f ə ɹ m eɪ ʃ ə n WORD_BOUNDARY d ʌ z n ɑ t WORD_BOUNDARY t̠ʃ eɪ n d̠ʒ WORD_BOUNDARY b eɪ s t WORD_BOUNDARY ɔ n WORD_BOUNDARY w ɪ t̠ʃ WORD_BOUNDARY ʌ v ð ə WORD_BOUNDARY t uː WORD_BOUNDARY ɹ æ n d ə m WORD_BOUNDARY v æ l j uː z WORD_BOUNDARY ɪ z WORD_BOUNDARY ɹ ɪ v iː l d WORD_BOUNDARY ð ɪ s WORD_BOUNDARY m iː n z WORD_BOUNDARY w iː WORD_BOUNDARY n oʊ WORD_BOUNDARY d̠ʒ ʌ s t WORD_BOUNDARY æ z WORD_BOUNDARY m ʌ t̠ʃ WORD_BOUNDARY ʌ b aʊ t WORD_BOUNDARY t ɛ m p ɹ ɪ t̠ʃ ə ɹ WORD_BOUNDARY w ɛ n WORD_BOUNDARY t oʊ l d WORD_BOUNDARY ð ə WORD_BOUNDARY m ʌ n θ WORD_BOUNDARY æ z WORD_BOUNDARY w iː WORD_BOUNDARY n oʊ WORD_BOUNDARY ʌ b aʊ t WORD_BOUNDARY ð ə WORD_BOUNDARY m ʌ n θ WORD_BOUNDARY w ɛ n WORD_BOUNDARY t oʊ l d WORD_BOUNDARY ð ə WORD_BOUNDARY t ɛ m p ɹ ɪ t̠ʃ ə ɹ WORD_BOUNDARY