JuniorTheoryCommon

What does this program print?

Python

p_spam = 0.1
p_word_spam = 0.8
p_word_ham = 0.2

spam = p_word_spam * p_spam
ham = p_word_ham * (1 - p_spam)
print(round(spam / (spam + ham), 2))

0.31

Expecting 0.8: that is P(word | spam); the code computes P(spam | word), and since spam is only 10% of mail, most emails with the word are normal.
Expecting 0.08: that numerator is P(word | spam) · P(spam); dividing by spam + ham, which is P(word) = 0.26, turns it into a probability.

How would the answer change if spam made up half of all mail?
Why can a classifier skip the division by P(word) when it only picks the class?

JuniorTheoryCommon

Python

p_spam = 0.1
p_word_spam = 0.8
p_word_ham = 0.2

spam = p_word_spam * p_spam
ham = p_word_ham * (1 - p_spam)
print(round(spam / (spam + ham), 2))

0.31

Expecting 0.8: that is P(word | spam); the code computes P(spam | word), and since spam is only 10% of mail, most emails with the word are normal.
Expecting 0.08: that numerator is P(word | spam) · P(spam); dividing by spam + ham, which is P(word) = 0.26, turns it into a probability.

How would the answer change if spam made up half of all mail?
Why can a classifier skip the division by P(word) when it only picks the class?