Editing Probability Theory (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Modeling 'Bayesian Inference' (Spam Filtering):'''
<syntaxhighlight lang="python">
def update_probability(prior_prob, prob_evidence_given_spam, prob_evidence_given_ham):
    """
    Bayes' Theorem: P(Spam|Word) = [P(Word|Spam) * P(Spam)] / P(Word)
    """
    # P(Word) = P(Word|Spam)P(Spam) + P(Word|Ham)P(Ham)
    p_ham = 1 - prior_prob
    p_word = (prob_evidence_given_spam * prior_prob) + (prob_evidence_given_ham * p_ham)
    
    posterior_prob = (prob_evidence_given_spam * prior_prob) / p_word
    return posterior_prob

# Prior: 10% of emails are spam. 
# 'Buy Now' appears in 80% of spam but only 1% of ham.
print(f"Prob it is spam if it says 'Buy Now': {update_probability(0.1, 0.8, 0.01):.2f}")
# This 'Learning' logic is how your email filter gets smarter.
</syntaxhighlight>

; Probability Paradoxes
: '''The Monty Hall Problem''' → Why you should always "switch doors" on a game show (it doubles your chances!).
: '''The Birthday Paradox''' → In a room of just 23 people, there is a 50% chance two share a birthday.
: '''Simpson's Paradox''' → A trend appears in several different groups of data but disappears or reverses when these groups are combined.
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