Discrete Probability Distributions
A discrete probability distribution is a an assignment of probabilities to the values of a discrete random variable. Discrete probability distributions arise from either empirical data or from mathematical formula. Below is an example of a discrete probability distribution created from empirical data.
A discrete probability distribution from empirical data.
Let the random variable x be the number of newspapers sold by a certain newsboy on a given day.
| x | p(x) |
| 0 | 0.10 |
| 1 | 0.12 |
| 2 | 0.35 |
| 3 | 0.20 |
| 4 | 0.15 |
| 5 | 0.08 |
From the table we see that the probability of the newsboy selling 2 papers is 0.35, and the probability that he will sell 4 papers is .15. It is necessary that the sum of all probabilities equal 1, otherwise the distribution is not a probability distribution.
A discrete probability distribution from mathematical formula
Let the random variable X be the number of heads obtained when one coin is tossed 3 times or 3 coins are tossed one time. From this experiment it is clear that the number of heads obtained could be none, one, two, or three.
X=[0,1,2,3]
Our next task is to assign probabilities to each value of X. Below is the sample space or the set of all possible outcomes when three coins are tossed.
| coin 1 | coin 2 | coin3 |
| H | H | H |
| H | H | T |
| H | T | H |
| H | T | T |
| T | H | H |
| T | H | T |
| T | T | H |
| T | T | T |
From the table we see that getting :
exactly 3 heads (HHH) occurs in 1 way.
exactly 2 heads (HHT),(HTH),(THH) occurs in 3 ways.
exactly 1 head (HTT) , (THT),)TTH) occurs in 3 ways.
exactly 0 (TTT) heads occurs in 1 way.
there are 8 possible outcomes.
Therefore:
Prob( HHH)= 1 way / 8 ways=1/8
Prob(HH) = 3 ways/8 ways=3/8
Prob(H)=3 ways/8 ways =3/8
Prob(~H)= 1 way/8ways =1/8
The probability distribution is presented below.
| x | p(x) |
| 0 | 1/8 |
| 1 | 3/8 |
| 2 | 3/8 |
| 3 | 1/8 |