Discrete Probability Distributions

Discrete Probability Functions also called Binomial Distribution assume a discrete number of values. For example, coin tosses and counts of events are discrete functions. These are discrete distributions because there are no in-between values. We can either have heads or tails in a coin toss.

For discrete probability distribution functions, each possible value has a non-zero probability. Moreover, the sum of all the values of probabilities must be one. For example, the probability of rolling a specific number on a die is 1/6. The total probability for all six values equals one. When we roll a die, we only get either one of these values.

Bernoulli Trials and Binomial Distributions

When we perform a random experiment either we get the desired event or we don’t. If we get the desired event then we call it a success and if we don’t it is a failure. Let’s say in the coin-tossing experiment if the occurrence of the head is considered a success, then the occurrence of the tail is a failure.

Each time we toss a coin or roll a die or perform any other experiment, we call it a trial. Now we know that in our experiments coin-tossing trial, the outcome of any trial is independent of the outcome of any other trial. In each of such trials, the probability of success or failure remains constant. Such independent trials that have only two outcomes usually referred to as ‘success’ or ‘failure’ are called Bernoulli Trials.

Definition:

Trials of the random experiment are known as Bernoulli Trials, if they are satisfying below given conditions :

• Finite number of trials are required.
• All trials must be independent.
• Every trial has two outcomes : success or failure.
• Probability of success remains  same in every trial.

Let’s take the example of an experiment in which we throw a die; throwing a die 50 times can be considered as a case of 50 Bernoulli trials, where the result of each trial is either success(let’s assume that getting an even number is a success) or failure( similarly, getting an odd number is failure) and the probability of success (p) is the same for all 50 throws. Obviously, the successive throws of the die are independent trials. If the die is fair and has six numbers 1 to 6 written on six faces, then p = 1/2 is the probability of success, and q = 1 – p =1/2 is the probability of failure.

Example: An urn contains 8 red balls and 10 black balls. We draw six balls from the urn successively. You have to tell whether or not the trials of drawing balls are Bernoulli trials when after each draw, the ball drawn is:

1. replaced
2. not replaced in the urn.

Answer:

1. We know that the number of trials are finite. When drawing is done with replacement, probability of success (say, red ball) is p =8/18 which will be same for all of the six trials. So, drawing of balls with replacements are Bernoulli trials.
2. If  drawing is done without replacement,  probability of success (i.e., red ball) in the first trial is 8/18 , in 2nd trial is 7/17 if  first ball drawn is red or, 10/18 if  first ball drawn is black, and so on. Clearly, probabilities of success are not same for all the trials, Therefore, the trials are not Bernoulli trials.

Binomial Distribution

It is a random variable that represents the number of successes in “N” successive independent trials of Bernoulli’s experiment. It is used in a plethora of instances including the number of heads in “N” coin flips, and so on. 

Let P and Q denote the success and failure of the Bernoulli Trial respectively. Let’s assume we are interested in finding different ways in which we have 1 success in all six trials.

Clearly, six cases are available as listed below:

PQQQQQ, QPQQQQ, QQPQQQ, QQQPQQ, QQQQPQ, QQQQQP

Likewise, 2 successes and 4 failures will show [Tex]\frac{6!}{4! 2!}  [/Tex] combinations thus making it difficult to list so many combinations. Henceforth, calculating probabilities of 0, 1, 2,…, n number of successes can be long and time-consuming. To avoid such lengthy calculations along with a listing of all possible cases, for probabilities of the number of successes in n-Bernoulli’s trials, a formula is made which is given as:

If Y is a Binomial Random Variable, we denote this Y∼ Bin(n, p), where p is the probability of success in a given trial, q is the probability of failure, Let ‘n’ be the total number of trials, and ‘x’ be the number of successes, the Probability Function P(Y) for Binomial Distribution is given as:

P(Y) = ⁿC_x q^n–xp^x 

where x = 0,1,2…n

Example: When a fair coin is tossed 10 times, find the probability of getting:

1. Exactly Six Heads
2. At least Six Heads

Answer: 

Every coin tossed can be considered as the Bernoulli trial. Suppose X is the number of heads in this experiment: 

We already know, n = 10

                            p = 1/2

So, P(X = x) = ⁿC_x p^n-x (1-p)^x , x= 0,1,2,3,….n

P(X = x) = ¹⁰C_xp^10-x(1-p)^x  

When x = 6, 

(i) P(x = 6) = ¹⁰C₆ p⁴ (1-p)⁶

                   = [Tex]\frac{10!}{6!4!}(\frac{1}{2})^{6}(\frac{1}{2})^{4}\\ \hspace{0.4cm} = \frac{7\times8\times9\times10}{2\times3\times4}\times\frac{1}{64}\times\frac{1}{16} \\ \hspace{0.4cm} = \frac{105}{512} [/Tex]

(ii) P(at least 6 heads) = P(X >= 6) = P(X = 6) + P(X=7) + P(X=8)+ P(X=9) + P(X=10) 

= ¹⁰C₆ p⁴ (1-p)⁶ + ¹⁰C₇ p³ (1-p)⁷ + ¹⁰C₈ p² (1-p)⁸ + ¹⁰C₉ p¹(1-p)⁹ + ¹⁰C₁₀ (1-p)¹⁰ =  

[Tex]\frac{10!}{6!4!}(\frac{1}{2})^{10} + \frac{10!}{7!3!}(\frac{1}{2})^{10} + \frac{10!}{8!2!}(\frac{1}{2})^{10} + \frac{10!}{9!1!}(\frac{1}{2})^{10} + \frac{10!}{10!}(\frac{1}{2})^{10}\\ \hspace{0.5cm} = (\frac{10!}{6!4!} + \frac{10!}{7!3!}+ \frac{10!}{8!2!} + \frac{10!}{9!1!}+ \frac{10!}{10!})(\frac{1}{2})^{10} \\ \hspace{0.5cm} = \frac{193}{512} [/Tex]

Negative Binomial Distribution

In a random experiment of discrete range, it is not necessary that we get success in every trial. If we perform ‘n’ number of trials and get success ‘r’ times where n>r, then our failure will be (n-r) times. The probability distribution of failure in this case will be called negative binomial distribution. For example, if we consider getting 6 in the die is success and we want 6 one time, but 6 is not obtained in the first trial then we keep throwing the die until we get 6. Suppose we get 6 in the sixth trial then the first 5 trials will be failures and if we plot the probability distribution of these failures then the plot so obtained will be called as negative binomial distribution.

Poisson Probability Distribution

The Probability Distribution of the frequency of occurrence of an event over a specific period is called Poisson Distribution. It tells how many times the event occurred over a specific period. It basically counts the number of successes and takes a value of the whole number i.e. (0,1,2…). It is expressed as

f(x; λ) = P(X=x) = (λ^xe^-λ)/x!

where, 

• x is number of times event occurred
• e = 2.718…
• λ is mean value

Binomial Distribution Examples

Binomial Distribution is used for the outcomes that are discrete in nature. Some of the examples where Binomial Distribution can be used are mentioned below:

• To find the number of good and defective items produced by a factory.
• To find the number of girls and boys studying in a school.
• To find out the negative or positive feedback on something

A probability distribution is an idealized frequency distribution. In statistics, a frequency distribution represents the number of occurrences of different outcomes in a dataset. It shows how often each different value appears within a dataset.

Probability distribution represents an abstract representation of the frequency distribution. While a frequency distribution pertains to a particular sample or dataset, detailing how often each potential value of a variable appears within it, the occurrence of each value in the sample is dictated by its probability.

A probability distribution, not only shows the frequencies of different outcomes but also assigns probabilities to each outcome. These probabilities indicate the likelihood of each outcome occurring.

In this article, we will learn what is probability distribution, types of probability distribution, probability distribution function, and formulas.