Conditional Probability and Independence

Conditional probability is the chance of an event after you already know that another event has happened. Extra information changes the odds.

Conditional Probability

P(A | B) = P(A and B) / P(B)

• P(A | B): the probability of A given that B has already happened. Read the bar as “given”.
• P(A and B): the probability that both events happen together.
• P(B): the probability of the event you know occurred. Dividing by it narrows the focus to only the cases where B is true.

Example: A company emails 200 customers. 80 opened the email, and 20 of those 80 made a purchase. Across all 200 customers, 40 made a purchase. Does opening the email change the chance of buying?

1. P(opened) = 80 / 200 = 0.40
2. P(opened and purchased) = 20 / 200 = 0.10
3. P(purchase | opened) = 0.10 / 0.40 = 0.25
4. P(purchase) overall = 40 / 200 = 0.20

Among people who opened the email, 25% bought, against 20% overall. Opening the email is linked to a higher chance of buying.

Independence

Two events are independent when knowing one tells you nothing about the other. Either of these conditions confirms it:

• P(A | B) = P(A)
• P(A and B) = P(A) × P(B)

In the example above, P(opened) × P(purchase) = 0.40 × 0.20 = 0.08, but P(opened and purchased) is 0.10. Since the two numbers differ, the events are not independent.