Conditional probability: Refresher

\(\mathbb P(\text{left} \mid \text{right})\) is the prob­a­bil­ity that the thing on the left is true as­sum­ing the thing on the right is true, and it’s defined as \(\frac{\mathbb P(\text{left} \land \text{right})}{\mathbb P(\text{right})}.\)

Thus, \(\mathbb P(yellow \mid banana)\) is the prob­a­bil­ity that a ba­nana is yel­low (“the prob­a­bil­ity of yel­low­ness given ba­nana”), while \(\mathbb P(banana \mid yellow)\) is the prob­a­bil­ity that a yel­low ob­ject is a ba­nana (“the prob­a­bil­ity of ba­nana, given yel­low­ness”).noteIn gen­eral, \(\mathbb P(v)\) is an ab­bre­vi­a­tion of \(\mathbb P(V = v)\) for some vari­able \(V\), which is as­sumed to be known from the con­text. For ex­am­ple, \(\mathbb P(yellow)\) might stand for \(\mathbb P({ColorOfNextObjectInBag}=yellow)\) where \(ColorOfNextObjectInBag\) is a vari­able in our prob­a­bil­ity dis­tri­bu­tion \(\mathbb P,\) and \(yellow\) is one of the val­ues that that vari­able can take on.

\(\mathbb P(x \land y)\) is used to de­note the prob­a­bil­ity of both \(x\) and \(y\) be­ing true si­mul­ta­neously (ac­cord­ing to some prob­a­bil­ity dis­tri­bu­tion \(\mathbb P\)). \(\mathbb P(x\mid y)\), pro­nounced “the con­di­tional prob­a­bil­ity of x, given y”, is defined to be the quantity

$$\frac{\mathbb P(x \wedge y)}{\mathbb P(y)}.$$

For ex­am­ple, in the Dise­a­sitis prob­lem, \(\mathbb P({sick}\mid {positive})\) is the prob­a­bil­ity that a pa­tient is sick given a pos­i­tive test re­sult, and it’s calcu­lated by tak­ing the 18% pa­tients who are sick and have pos­i­tive test re­sults, and di­vid­ing by all 42% of the pa­tients who got pos­i­tive test re­sults. That is, \(\mathbb P({sick}\mid {positive})\) \(=\) \(\frac{\mathbb P({sick} \land {positive})}{\mathbb P({positive})}.\)

Us­ing a fre­quency di­a­gram, we can vi­su­al­ize \(\mathbb P(sick \mid positive)\) as the prob­a­bil­ity of draw­ing a \(sick\) re­sult from a bag of only those peo­ple in the pop­u­la­tion who got a \(positive\) re­sult.

diseasitis frequency

bag of 18 and 24 patients

The “given” op­er­a­tor in \(\mathbb P(x\mid y)\) tells us to as­sume that \(y\) is true, to re­strict our at­ten­tion to only pos­si­ble cases where \(y\) is true, and then ask about the prob­a­bil­ity of \(x\) within those cases.

Note that \(\mathbb P(positive \mid sick)\) is not the same as \(\mathbb P(sick \mid positive).\) To find the prob­a­bil­ity that a pa­tient has a pos­i­tive re­sult given that they’re sick, we can vi­su­al­ize tak­ing the 20 sick pa­tients and putting them in a group, and then ask­ing the prob­a­bil­ity that a ran­domly se­lected one will have a pos­i­tive re­sult, which will be \(\frac{18}{20} = 0.9\) — so \(\mathbb P(positive \mid sick) = 90\%,\) while \(\mathbb P(sick \mid positive) \approx 43\%.\) Mix­ing up which one is which is an un­for­tu­nate source of of many prac­ti­cal er­rors when you’re try­ing to do these calcu­la­tions us­ing only the for­mal no­ta­tion, at least un­til you get used to it. Just re­mem­ber that \(\mathbb P(\text{left} \mid \text{right})\) is the prob­a­bil­ity of the thing on the left given that the thing on the right is true.


  • Conditional probability

    The no­ta­tion for writ­ing “The prob­a­bil­ity that some­one has green eyes, if we know that they have red hair.”