Statistics Conditional Probability

What Is Conditional Probability?

Conditional probability answers a very specific question: given that we already know something happened, how likely is another event? The plain probability that a random person owns a winter coat is different from the probability that someone living in a cold climate owns one — new information shifts the odds. We write this as P(A|B), read as 'the probability of A given B'.

The Conditional Probability Formula

The formula is P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability that both A and B happen, and P(B) is the probability that B happens at all. This only makes sense when P(B) is greater than zero — you cannot condition on an event that never occurs. In plain terms: out of all the times B happens, what fraction of those times does A also happen?

total_cards = 52
kings = 4
face_cards = 12  # Jacks, Queens, Kings: 4 of each

p_a_and_b = kings / total_cards   # P(King AND Face card) = P(King), since every king is a face card
p_b = face_cards / total_cards    # P(Face card)

p_a_given_b = p_a_and_b / p_b
print(f"P(King | Face card) = {p_a_given_b:.4f}")  # 0.3333

• P(Exercises) = 120/200 = 0.60
• P(Sick) = 50/200 = 0.25
• P(Sick AND Exercises) = 20/200 = 0.10
• P(Sick | Exercises) = 0.10 / 0.60 ≈ 0.167 — noticeably lower than the overall 25%, which hints that exercise and sickness are related

exercises_and_sick = 20
exercises_total = 120
overall_sick = 50
population = 200

p_sick_given_exercise = exercises_and_sick / exercises_total
p_sick_overall = overall_sick / population

print(f"P(Sick | Exercises) = {p_sick_given_exercise:.3f}")  # 0.167
print(f"P(Sick) overall     = {p_sick_overall:.3f}")        # 0.250

Independent vs Dependent Events

Two events are independent when knowing one tells you nothing new about the other — mathematically, P(A|B) = P(A). If the conditional probability differs from the plain probability, the events are dependent: one influences the likelihood of the other, as we just saw with exercise and sickness.

outcomes = list(range(1, 7))  # a fair six-sided die

event_a = {2, 4, 6}   # rolling an even number
event_b = {5, 6}      # rolling a number greater than 4

p_a = len(event_a) / len(outcomes)
p_b = len(event_b) / len(outcomes)
p_a_and_b = len(event_a & event_b) / len(outcomes)

p_a_given_b = p_a_and_b / p_b

print(f"P(A) = {p_a:.3f}, P(A|B) = {p_a_given_b:.3f}")  # both 0.500 -> independent