doing math

courses › Business Stats & Decisions

Base Rates & Conditional Rates

level 37 course

Always ask 'out of which group?' before you divide.

In your head, jot if needed · no calculator why?

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Builds on: Probability Basics (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

Every rate is a count divided by a group, and the group decides the answer. Before you divide, ask "out of which group?"

"The conversion rate among referrals" divides by referrals. "The share of buyers who were referrals" divides by buyers. Same counts, different questions, different answers.

The biggest trap is a warning sign for something rare. If 2% of orders are fraud, many flags can land on honest orders, because there are so many honest ones to be wrong about.

Techniques

Name the group, then divide

Any rate or share: among a group, or out of all buyers.

  1. Find the group after "among" or "of all". It goes on the bottom.
  2. On top, count only members of that group with the outcome.
  3. Say it back in words before you divide.
worked example

Example: Referrals: 30 buy, 20 don't. Non-referrals: 10 buy, 40 don't. What share of all buyers were referrals?

  1. The group is buyers: 30 + 10 = 40.
  2. 30 ÷ 40 = 0.75 = 75%.
  3. A different question: among referrals, 30 ÷ 50 = 60% buy.

Answer: 75%

Turn rates into counts

A filter flags something rare, and you want the chance a flag is real.

  1. Split the population into the rare group and the rest.
  2. Apply each flag rate to its own group.
  3. Real share of flags: flagged rare ÷ all flagged.
worked example

Example: Of 1,000 orders, 20 are fraudulent. A filter flags 90% of fraudulent orders and 5% of honest orders. If an order is flagged, what is the chance it's fraudulent? Round to the nearest whole percent.

  1. Flagged fraud: 90% of 20 = 18.
  2. Flagged honest: 5% of 980 = 49.
  3. 18 ÷ 67 ≈ 26.9%, about 27%.

Answer: 27%

Tips by skill

  • TipRate within a group: Divide by the group after "among", not by everyone. On top go the ones in that group with the outcome.
  • TipShare of buyers from a group: The group is now buyers: this group's buyers ÷ all buyers.
  • TipFlagged: how likely is it real?: Turn the rates into counts, flagged fraud and flagged honest. Divide flagged fraud by all flagged orders.
  • TipWhich denominator?: The whole group named after "among" goes on the bottom. Only those with the outcome go on top.

Watch out for

  • Dividing by everyone instead of the group named. "Among referrals" means divide by referrals, not all leads.
  • Answering the rate within a group when the question asks for a share of buyers, or the other way round.
  • Confusing "flagged, given fraud" with "fraud, given flagged". A filter that catches 90% of fraud can still be wrong about most of its flags.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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