Define and calculate conditional probabilities

Given two events \(A\) and \(B\), the conditional probability of event \(A\) occurring, given that event \(B\) has occurred is the probability of event \(A\) and \(B\) occurring over the probability of event \(B\) occurring as shown in the formula below:

$$ P\left( A|B \right) =\frac { P\left( A\cap B \right) }{ P\left( B \right) } $$
as long as \(P (B) > 0\).

As shown in the diagram below, the probability of \(A\) given \(B\) occurred is the shaded region \(A \cap B\).

Example

Suppose 50% of the students in a school take geometry, 30% take history and 15% take both geometry and history. Given that a student selected at random takes geometry, what is the probability he or she also takes history.

Let event \(A\) be “takes geometry” and event \(B\) be “takes history”.

\(P \left(B \right) = .30\)

\(P \left(A \cap B \right) = .15\)

$$ P\left( A|B \right) =\frac { P\left( A\cap B \right) }{ P\left( B \right) } ={ .15 }/{ .30 }=.50\quad or\quad 50\% $$

This concept can be extended to more than 2 events as shown in the formula for 3 events below:

$$ P\left( C|A\cap B \right) =\frac { P\left( A\cap B\cap C \right) }{ P\left( A \right) \ast P\left( B|A \right) } $$

Example

Suppose of students at a university:

 45% take chemistry

 35% take biology

 25% take math

 10% take chemistry and math

 5% take chemistry and biology

 3% take biology and math

 1% take all three subjects

Calculate the probability that a student takes math, given that they take chemistry and biology.

\(P\left( Math|Chemistry\quad and\quad Biology \right) ={ P\left( Math\quad and\quad Chemistry\quad and\quad Biology \right) }/{ P\left( Chemistry \right) \ast P\left( Biology|Chemistry \right) }\)

\(\begin{align*} P\left( Biology|Chemistry \right) & ={ P\left( Biology\quad and\quad Chemistry \right) }/{ P\left( Chemistry \right) } \\ & ={ .05 }/{ .45 }={ 1 }/{ 9 } \\ \end{align*} \)

\(P(Math|Chemistry\quad and\quad Biology)={ .01 }/{ \left( { .45*.05 }/{ .45 } \right) }={ .01 }/{ .05 }={ 1 }/{ 5 }=.20\quad or\quad 20\%\)

 

Learning Outcome

Topic 1.e: General Probability – Define and calculate conditional probabilities.


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