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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:

Probability distributions are of two types:

1. Discrete Probability Distribution

2. Continuous Probability Distribution or Probability Density Function

A discrete probability distribution 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 continuous probability distribution ( or probability density function) 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.]

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:

$\begin{array}{l}P\left(X=0\right)={}_5{C}_0\cdot {\left(\frac{3}{4}\right)}^5\\ =1\cdot \frac{243}{1024}\end{array}$

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:

$\begin{array}{l}P\left(X=1\right)={}_5{C}_1\cdot \left(\frac{1}{4}\right)\cdot {\left(\frac{3}{4}\right)}^4\\ =5\cdot \frac{81}{1024}\\ =\frac{405}{1024}\end{array}$

(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.