Runouts and outs

Use complements for multiple draws while keeping “improves” separate from “wins.”

10 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Compute at-least-one probabilities with fixed outs and explain the model’s limits.

Prerequisites: Complements and unions · Conditional draws

An out is defined by the event

An out is a card that achieves the specified result. Completing a flush is not automatically the same event as winning at showdown. An opponent may make a stronger hand or a later card may change the result. State whether outs mean improvement or clean winners.

With 47 unseen cards and 9 fixed outs, hitting on the next card has probability 9/47. For at least one hit in two draws, calculate the complement of two misses. After a miss there are 46 unseen cards and 37 misses.

At least one is not exactly one

The two-draw probability is 1−(38/47)(37/46), about 34.968%. It includes hit-hit as well as the two one-hit orderings. Adding the two marginal hit probabilities would count some runouts twice.

This calculation assumes uniform draws and a fixed designated out set. It says nothing about future betting cost or whether you can realize the full two-card probability before facing another decision. A precise runout probability is only one input to a decision model.

Worked example

A 10-card toy deck contains 3 designated outs. Draw two without replacement. What is P(at least one out)?

Seven cards miss on the first draw.
Given a miss, six of the remaining nine cards miss again.
Compute 1−(7/10)(6/9)=1−42/90.

The chance is 48/90=8/15, about 53.33%. The at-least-one event includes both outs appearing.

Common mistakes

Equating hand improvement with guaranteed victory.
Adding draw probabilities with overlapping outcomes.
Using a two-card probability to price a decision that buys only one card.

Keep these ideas

Define the event that makes a card an out.
Use one minus all misses for a fixed out set.
Keep runout mathematics separate from the betting model.