Multiplication of Probabilities

Probability → Vocabulary → Formula → Example Questions → Addition of Probabilities → Multiplication of Probabilities

Consider this example, a dice is rolled once, what is the probability that the number shown is Even and is four?

$P(A\textrm{ and }B)=P(A\cap B)=P(A)\cdot P(B)$

So, this is the multiplication of probabilities.

Solution to the above problem
A dice is rolled, so all possible events are six {1,2,3,4,5,6}, even numbers are three {1,3,5}, and four is only one event {4}. Thus,

$P(\textrm{even})=\frac{3}{6}=0.5$ , and

$P(4)=\frac{1}{6}=0.1667$

Hence,

$P(\textrm{even and }4)=P(\textrm{even})\cdot P(4)=0.5\times 0.1667=0.0833$

SO REMEMBER:

$P(A\textrm{ or }B)=P(A\cup B)=P(A)+P(B )$ for disjoint events
$P(A\textrm{ or }B)=P(A\cup B)=P(A)+P(B )-P(A\cap B)$ for joint events
$P(A\textrm{ and }B)=P(A\cap B)=P(A)\cdot P(B )$ for joint events
$P(A\textrm{ and }B)=P(A\cap B)=0$ for disjoint events (for example, the probability that the number is even and 3 is 0, because it cannot happen).