Checking whether events are independent

The common mistake

Multiplying P(A) × P(B) for events that affect each other, such as drawing two cards without putting the first one back.

Instead: Ask: does knowing A happened change the chance of B? If it does, use the conditional probability P(B | A) for the second step. To test independence from data, check whether P(A and B) equals P(A) × P(B), or whether P(B | A) equals P(B).

Worked example with Guide me

Two cards are drawn from a 52-card deck without replacement. What is P(both are aces)?

1. Sugar · What's the chance the first card is an ace?
   Student · 4/52.
2. Sugar · After one ace is gone, does the second chance change?
   Student · Yes — 3 aces left out of 51, so 3/51.
3. Sugar · Now multiply those.
   Student · 4/52 × 3/51 = 12/2652.

Answer: 1/221 — not (4/52)², because the draws aren't independent.

Quick answers

What is the most common mistake with checking whether events are independent?

Multiplying P(A) × P(B) for events that affect each other, such as drawing two cards without putting the first one back.

How do I get checking whether events are independent right?

Ask: does knowing A happened change the chance of B? If it does, use the conditional probability P(B | A) for the second step. To test independence from data, check whether P(A and B) equals P(A) × P(B), or whether P(B | A) equals P(B).

Can you show a worked example?

Two cards are drawn from a 52-card deck without replacement. What is P(both are aces)? 1/221 — not (4/52)², because the draws aren't independent.

Related concepts

• Multiplying probabilities for “and” events
• Cumulative frequency graphs