Conditional directions

Distinguish how often a hand produces an action from how often that action means the hand is strong.

10 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Read the direction of a conditional probability and identify its denominator.

The information after the bar defines the population

Write S for a strong hand and B for a bet. P(B|S) asks for the fraction of strong-hand cases in which the player bets. P(S|B) instead asks for the fraction of betting cases in which the player is strong. The conditioning event after the bar changes which trials you count.

These two probabilities can be very different even when computed from the same table. Reversing a statement without changing its denominator is an inference error. Naming the event as “evidence given state” or “state given evidence” often makes the direction clear.

Use the joint event as the bridge

Both conditionals involve the same joint event, S and B. P(B|S)=P(S and B)/P(S), while P(S|B)=P(S and B)/P(B), when the denominators are positive. The shared numerator does not make the ratios equal.

A behavioral likelihood is a model of how a state generates an observed action. To turn it into a posterior belief about the state, you also need how common that state was and how often alternatives would produce the action.

Worked example

In 100 modeled hands, 20 are strong. The player bets 16 strong hands and 24 weak hands. Compare both conditional directions.

Among strong hands, 16 of 20 bet, so P(B|S)=80%.
There are 16+24=40 bets total.
Among bets, 16 of 40 are strong, so P(S|B)=40%.

80% describes betting behavior when strong; 40% describes strength after a bet. Neither can replace the other.

Common mistakes

Reading the conditional bar as a symmetric relationship.
Using all hands as the denominator after conditioning on a bet.
Assuming a high likelihood makes the posterior equally high.

Keep these ideas

Read the conditioning event first.
The denominator is the population allowed by the given information.
A likelihood is not automatically a posterior.