In this video we discuss what a discrete probability is and how to construct a discrete probability distribution. We also cover how to graph a discrete probability distribution and go through a full example. Transcript/notes
Discrete probability distribution
We first need to start with variables. A random variable is a variable that is determined by chance, and there are discrete and continuous variables.
Discrete variables can be counted using whole numbers, 0, 1, 2, 3, and so on, for instance, the number of pitches thrown by a pitcher in a baseball game or the number of people in a restaurant.
Continuous variables have an infinite number of values, so they can’t be counted, variables like heights, weights, time and temperatures are continuous variables.
A discrete probability distribution lists the values a random variable can assume, and lists the probabilities for those values. For instance, here is a discrete probability distribution for rolling a single die. The possible values, 1 through 6, for the random variable x are listed in the top column and the second column lists the probabilities for each of the possible values.
Probability distributions can also be shown graphically.
As an example, let’s say you created a new fitness drink and at a recent expo you sold 6 four packs, 15 eight packs, 9 12 packs and 3 24 packs as you see here in this frequency distribution table. From this data we can create a probability distribution. The top row is the variable x, which is the 4 different types of packs, 4 pack, 8 pack, 12 pack and 24 pack. And the bottom row is the probability of the variable x, which we need to calculate from the frequency table. First we need to tally up the total frequency or total number of units sold, which is 33.
For the 4 pack, or probability of 4, we sold 6 units, divide that by the total of 33, and rounding off we get 0.18. So, the probability of selling a 4 pack is 18% or 0.18, and we put that in column 2 under the 4.
For the 8 pack, or probability of 8, we sold 15 units, divide that by the total of 33, and rounding off we get 0.45. So, the probability of selling an 8 pack is 45% or 0.45, and we put that in column 2 under the 8.
And we will use this same process for the 12 pack, which has a probability of 0.27 and for the 24 pack which ends up with a probability of 0.09. And we put these into the probability distribution.
Now that our probability distribution is complete, we can graph the data, starting with drawing an x and y axis. We can label the x axis as types of drink packs and the y axis is going to represent the probability, P of x. Next we are going to mark the y axis and from our probability distribution table we find the highest probability, which is 0.45 here. So, I am going to mark that near the top of the y axis here. And then I’m going to go down the y axis marking increments of 5, making sure the marks are equidistant from one another like this.
Next we are going to mark the x axis with 4 marks, again equidistant from one another, and label the marks, 4 pack, 8 pack, 12 pack and 24 pack. Now we can draw in the bars. For the four pack, we have a probability of 0.18, so we find that on the y axis, and above the 4 pack mark on the x axis we draw in a horizontal line and then connect it down to the x axis and fill in the bar. Have the right side of the bar be midway between the 4 pack and 8 pack marks on the x axis and the bar should be centered around the 4 pack mark.
Now for our next bar, for the 8 pack, which has a probability of 0.45. Again, find 0.45 on the y axis, draw in a line above the 8 pack on the x axis, and connect it down to the x axis and fill in the bar. Again, this bar should end between the 8 pack mark and the 12 pack mark on the x axis, to keep the graph looking clean, balanced and neat.
Then, use this same process for the remaining 2 items, the 12 pack, which has a probability of 0.27, and the 24 pack with a probability of 0.09. And our histogram is complete.
One note with this graph, each of these bars represents a width of 1, and the total area of all the bars is equal to 1. So, for instance, what is the probability of selling a 4 pack or a 24 pack?
The answer to this can be looked at as an area problem, the area of the 4 pack is width times height, 1 times 0.18, and the area of the 24 pack is 1 times 0.09. Add these together, 0.18 plus 0.09 equals 0.27. So, the probability of selling a 4 pack or a 24 pack is 0.27 or 27%. This is the same principle as the addition rule for probability as you see here. And this is a very, very important concept that will be used in later videos. Just remember this graph represents an area.
One last thing to cover is that there are 2 requirements for probability distributions. First, the probability for each of the possible values for the random variable is between 0 and 1, including 0 and 1. Second, the sum of all the probabilities must equal 1.
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