Dependent Events - Examples

# Dependent Events

When something has happened we say an event is occurred. All events are associated to a random experiment. For example: throwing a dice is an experiment while getting a 5 on a dice is event associated to that experiment.
Depending on the various conditions we classify events as dependent or independent:

The dependent events are the ones in which the occurrence or outcome of the first event is affecting the occurrence or outcome of the next event in line. For example if we draw two cards from a given deck of 52 cards, then the event of getting a heart first and then getting a red queen are dependent events.

The independent events are the ones in which occurrence of the first event does not at all affect the occurrence of the next event in line. Likewise, if we take up the same example above on the condition that after the first draw the card is replaced back before the next draw then the same events will be independent of each other.
For finding the probability of independent events we simply multiply the probabilities, while for finding the probability of dependent events we either find the probabilities before multiplying by good analysis or by using the conditional probability.

## Dependent Events Examples

Let us see examples on both.
Example 1:

Two cards have been drawn from the deck of 52 cards without replacing the first one back. Find the probability of getting first card as king and second card as queen.

Solution:

Clearly, the two events are dependent.

Let A be the event of drawing the king first, so $P (A)$ = $\frac{4}{52}$

Now one card is drawn already so we are left with 51 cards only.

Let B be the event of drawing a queen next, so $P (B)$ = $\frac{4}{51}$

Now the compound probability is given by:

$P (A\ and\ B)$ = $P (A) . P (B)$ = $\frac{4}{52}$ . $\frac{4}{51}$ = $\frac{16}{2652}$ = $\frac{4}{663}$

So the probability of choosing a king and then a queen is $\frac{4}{663}$.
Example 2:

In a certain test 5 out of 20 students scored an ‘A’. We chose three students at random out of the 20 students without replacement. Find the probability that all three are the ones who scored an ‘A’.

Solution:

It is clear that all three events are dependent events.

Let A be the event of choosing first student with grade $‘A’$, so $P (A)$ = $\frac{5}{20}$

Now number of students is equal to 19 and number of students with grade ‘A’ are 4.

Let B be the event of choosing second student with grade $‘A’$, so $P (B)$ = $\frac{4}{19}$

Now number of students is equal to 18 and number of students with grade ‘A’ are 3.

Let C be the event of choosing third student with grade $‘A’$, so $P (C)$ = $\frac{3}{18}$

Hence the compound probability of all three is given by:

$P (A\ and\ B\ and\ C)$ = $P (A) P (B) P (C)$ = $\frac{5}{20}$ . $\frac{4}{19}$ . $\frac{3}{18}$ = $\frac{1}{114}$.

Hence the probability of choosing all three students with grade ‘A’ is $\frac{1}{114}$.