# Probability Distribution

A probability distribution for a particular random variable is a function or table of values that maps the outcomes in the sample space to the probabilities of those outcomes.

For example, in an experiment of tossing a coin twice, the sample space is

{HH, HT, TH, TT}.

Here, the random variable , $X$ , which represents the number of tails when a coin is tossed twice, takes the values $0,1$ and $2$ . The probability distribution for this experiment is given in table form:

Number of Trials( $X$ ) | $0$ | $1$ | $2$ |

Probability | $\frac{1}{4}$ | $\frac{1}{2}$ | $\frac{1}{4}$ |

Probability distributions are of two types:

1. Discrete Probability Distribution

2. Continuous Probability Distribution or Probability Density Function

A
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discrete probability distribution
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is one which lists the probabilities of random values with integer type or countable values.

[The binomial probability distribution is an example of a discrete probability distribution.]

A
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continuous probability distribution (
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or
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probability density function)
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is one which lists the probabilities of random variables with values within a range and is continuous.

[The normal probability distribution is an example of a continuous probability distribution. There are others, which are discussed in more advanced classes.]

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Example:
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Suppose you take a multiple-choice test with five questions, where each question has four choices, and you guess randomly on each question. Find the probability distribution for the number of correct answers.

For each problem, there are four choices and only one correct choice. So, the probability of getting a correct answer is $\frac{1}{4}$ and probability of getting an incorrect answer is $\frac{3}{4}$ .

Let the random variable $X$ be the number of correct answers. It can take the values $0,1,2,3,4$ and $5$ . This is an example of a binomial probability distribution , where $n=5$ , $p=\frac{1}{4}$ and $q=\frac{3}{4}$ .

Find $P\left(X=0\right)$ .

Here, all five guesses are incorrect.

The probability of getting zero correct answers is:

$\begin{array}{l}P\left(X=0\right)={}_{5}{C}_{0}\cdot {\left(\frac{3}{4}\right)}^{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1\cdot \frac{243}{1024}\end{array}$

Find $P\left(X=1\right)$ .

Here, one guess is correct and the other four guesses are incorrect.

The probability of getting one only correct answer is:

$\begin{array}{l}P\left(X=1\right)={}_{5}{C}_{1}\cdot \left(\frac{1}{4}\right)\cdot {\left(\frac{3}{4}\right)}^{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=5\cdot \frac{81}{1024}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{405}{1024}\end{array}$

(Note that we had to use combinations here. There is only one way to get zero questions correct, but $5$ different ways to get exactly one question correct. There are ${}_{5}{C}_{2}=10$ different ways to get exactly two questions correct.)

Similarly, find the remaining probabilities and make the table of probability distribution.

$X$ = Number of Correct Answers | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ |

Probability | $\frac{243}{1024}$ | $\frac{405}{1024}$ | $\frac{270}{1024}$ | $\frac{90}{1024}$ | $\frac{15}{1024}$ | $\frac{1}{1024}$ |