# Probability

**
Probability
**
is a measure of how likely something is to happen. It is expressed with a number between
$0$
and
$1$
;
$0$
means "impossible" and
$1$
means "certain".

For example: suppose you are asked to randomly choose a letter of the alphabet between $J$ and $N$ , inclusive. The possibilities are $J,K,L,M,N$ . The probability that you pick a vowel is $0$ . The probability that you pick a consonant is $1$ . The probability that you pick $K$ is $\frac{1}{5}=0.2$ .

In general, the probability that an event occurs is given by:

$P\left(\text{event}\right)=\frac{\text{numberoffavorableoutcomes}}{\text{numberofpossibleoutcomes}}$

**
Example:
**

Suppose you roll a six-sided die. Find the probability that you roll a number greater than four.

There are six possible outcomes: $1,2,3,4,5$ and $6$ . There are two favorable outcomes: $5$ and $6$ . So the probability is:

$P\left(\text{greaterthan}4\right)=\frac{2}{6}=\frac{1}{3}$

This is a
**
theoretical probability
**
, as opposed to
**
experimental probability
**
, which is the
*
observed
*
number of favorable outcomes out of a certain number of trials. For instance, suppose you rolled the six-sided die
$5$
times, and got the following results:
$2$
,
$6$
,
$4$
,
$5$
,
$6$
. Then the
*
experimental probability
*
of rolling a number greater than four would be
$\frac{3}{6}$
or one-half.

As the number of trials increases, the experimental probability will become close to the theoretical probability.

Calculating probabilities of more than one event can be challenging. Sometime we add the probabilities of each event, and sometime we multiply them. We can use key words (e.g., or, and, etc.) to identify the operations to use. That is, if you have the key words, “either”, “or”, “at least” or their synonyms, you need to add the probabilities of each independent events to find the combined probability. When you have the key words such as “and”, “both”, “all” or their synonyms, you need to multiply the probabilities of each independent event to find the combined probability.

For example, when a fair die is thrown, what is the probability of getting a $1$ or a $4$ ? Here, there are $6$ possible outcomes and probability of getting the number $1$ or $4$ is $\frac{1}{6}$ each. So, the probability of getting a $1$ or $4$ is $\frac{1}{6}+\frac{1}{6}$ or $\frac{2}{6}=\frac{1}{3}$ .

Now consider choosing a white first and then a green ball from a box of $4$ black, $7$ green and $9$ white identical balls if you replace the ball after the first draw.

There are $9$ white balls out of $20$ identical balls in the box. So, the probability of choosing a white ball first is $\frac{9}{20}$ . The ball is replaced after the first draw, so the total number of balls remains the same and there are $7$ green balls in the box. So, the probability of choosing a green ball is $\frac{7}{20}$ . Now, to find the probability of choosing a white first and then a green ball, you need to multiply the probabilities of the two events. Therefore, the probability of the combined event is $\frac{9}{20}\cdot \frac{7}{20}$ or $\frac{63}{400}$ .

You can also check addition rule in probability and multiplication rule in probability .