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.