# Compound Probability

The outcomes of a random experiment are known as events. There can more than one number of events associated with an experiment. A compound event is consisting of two or more number of simple events. Likewise: rolling two fair dice together.

Compound probability is the probability of joint occurrence of two or more simple events. Either all events can be true at the same time or either of them can be true at a time. When at least one of the events holds true at a time in a compound event then we say that the events are mutually exclusive, else they are non-mutually exclusive.In case of non-mutually exclusive events we have two types of events: independent and dependent. Independent events are the ones whose occurrence does not affect another. Else the events are termed as dependent events.

## Formula

In the cases when the events are independent, the compound probability can be calculated by multiplying the probability of the two events directly.

Let X and Y be two independent events. Then,

P (X and Y) = P (X) * P (Y)

When the events are dependent then the compound probability is given by:

P (X and Y) = P (X) * P(Y following X)

These are the cases when both parts of the compound events are true.

When one or more part of the compound event holds true, then the compound probability is given by:

P (X or Y) = P (X) + P (Y)
The word ‘and’ means all parts of the compound events are true at the same time while the word ‘or’ indicates that either or both of the parts of the compound events are true at a time. Basically when the events are mutually exclusive we use ‘or’ and when the events are not mutually exclusive we use ‘and’.

## Problems

Example 1: A basket has 12 marigolds and 8 carnations. If we had picked two flowers one by one with replacement, find the probability of getting a marigold and a carnation.

Solution:

Let $S$ be the same space, then $n (S)$ = 20

$X$, the event of picking a marigold, $n (X)$ = 12

$Y$, the event of picking a carnation, $n (Y)$ = 8

Then we have, $P (X)$ = $\frac{12}{20}$ and $P (Y)$ = $\frac{8}{20}$

Since the events are independent, then the compound probability is given by the product of the two probabilities.

$\rightarrow$ $P (X\ and\ Y)$ = $P (X) . P (Y)$ = $\frac{12}{20}$ * $\frac{8}{20}$ = $\frac{96}{400}$ = $\frac{6}{25}$

Example 2: An urn has 6 black and 4 white marbles. Two of them are drawn without replacement. Find the probability of getting a black and a white marble.

Solution:

Let S be the sample space. Then, $n (S)$ = 10

$X$, event of drawing a black marble, $n (X)$ = 6

$Y$, event of drawing a white marble, $n (Y)$ = 4

$P (X)$ = $\frac{6}{10}$ = $\frac{3}{5}$

$P (Y)$ = $\frac{4}{10}$ = $\frac{2}{5}$

$P (Y|X)$ = probability of getting white marble after drawing black = $\frac{4}{9}$

Since the events are not dependent, we have,

$P (X\ and\ Y)$ = $P (X) . P (Y|X)$ = $\frac{3}{5}$ * $\frac{4}{9}$ = $\frac{12}{45}$ = $\frac{4}{15}$.