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# Define and calculate conditional probabilities

## Conditional Probability

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 by 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: Conditional Probability

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?

Solution

Let event $$A$$ be “takes geometry” and event $$B$$ be “takes history.”

$$P \left(B \right) = 0.30$$

$$P \left(A \cap B \right) = 0.15$$

$$P\left( B|A \right) =\frac { P\left( A\cap B \right) }{ P\left( A \right) } ={ 0.15 }/{ 0.50 }=0.30\quad or\quad 30\%$$

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

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

#### Example: Conditional Probability

Suppose that the students at a university are grouped such that:

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.

$$P\left( \text{Math|Chemistry and Biology} \right) =\frac{{ P\left( \text{Math and Chemistry and Biology} \right) }}{{ P\left( \text{Chemistry} \right) \times P\left( \text{Biology|Chemistry} \right) }}$$

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

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

The definition of conditional probability can be rewritten to come up with multiplication rule for probabilities:

$$P(A\cap B)=P(A|B)\bullet P(B)$$

Also, if the events A and B are independent, then:

$$P(A|B)=P(A)$$

Learning Outcome

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

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