Multiplication Rule in Probability

If $A$ and $B$ are two independent events in a probability experiment, then the probability that both events occur simultaneously is:

$P\left(A\text{ and }B\right)=P\left(A\right)\cdot P\left(B\right)$

In case of dependent events , the probability that both events occur simultaneously is:

$P\left(A\text{ and }B\right)=P\left(A\right)\cdot P\left(B|A\right)$

(The notation $P\left(B|A\right)$ means "the probability of $B$ , given that $A$ has happened.")

The two events are independent events; the choice of hat has no effect on the choice of shirt.

There are three different hats, so the probability of choosing the songkok is $\frac{1}{3}$ . There are four different shirts, so the probability of choosing the black shirt is $\frac{1}{4}$ .

So, by the Multiplication Rule:

$\begin{array}{l}P\left(\text{songok and black shirt}\right)=\frac{1}{3}\cdot \frac{1}{4}\\ =\frac{1}{12}\end{array}$

The two events are dependent events because the first card is not replaced.

There is only one ace of spades in a deck of $52$ cards. So:

$P\left({\text{1}}^{\text{st}}\text{ card is the ace of spades}\right)=\frac{1}{52}$

If the ace of spaces is drawn first, then there are $51$ cards left in the deck, of which $13$ are hearts:

$P\left(2^{\text{nd}}{\text{ card is a heart | 1}}^{\text{st}}\text{ cardis the ace of spades}\right)=\frac{13}{51}$

So, by the multiplication rule of probability, we have:

$\begin{array}{l}P\left(\text{ace of spades, then a heart}\right)=\frac{1}{52}\cdot \frac{13}{51}\\ =\frac{13}{4\cdot 13\cdot 51}\\ =\frac{1}{204}\end{array}$