Probability Mass Functions
For a random variable, the most important thing to know is: how likely is each outcome? For a discrete random variable, this information is called the "probability mass function". The probability mass function (PMF) provides the "mass" (i.e. amount) of "probability" for each possible assignment of the random variable.
Formally, the probability mass function is a mapping between the values that the random variable could take on and the probability of the random variable taking on said value. In mathematics, we call these associations functions. There are many different ways of representing functions: you can write an equation, you can make a graph, you can even store many samples in a list. Let's start by looking at PMFs as graphs where the
In the following example, on the left we show a PMF, as a graph, for the random variable:


The sum of two dice example in the equally likely probability section. Again, the information that is provided in these graphs is the likelihood of a random variable taking on different values. In the graph on the right, the value "
Here is the exact same information in equation form:
As a final example, here is the PMF for
def pmf_sum_two_dice(y):
# Returns the probability that the sum of two dice is y
if y < 2 or y > 12:
return 0
if y <= 7:
return (y-1) / 36
else:
return (13-y) / 36
Notation
You may feel that
Probabilities Must Sum to 1
For a variable (call it
For further understanding, let's derive why this is the case. A random variable taking on a value is an event (for example
Data to Histograms to Probability Mass Functions
One surprising way to store a likelihood function (recall that a PMF is the name of the likelihood function for discrete random variables) is simply a list of data. We simulated summing two die 10,000 times to make this example dataset:
[8, 4, 9, 7, 7, 7, 7, 5, 6, 8, 11, 5, 7, 7, 7, 6, 7, 8, 8, 9, 9, 4, 6, 7, 10, 12, 6, 7, 8, 9, 3, 7, 4, 9, 2, 8, 5, 8, 9, 6, 8, 7, 10, 7, 6, 7, 7, 5, 4, 6, 9, 5, 7, 4, 2, 11, 10, 11, 8, 4, 11, 9, 7, 10, 12, 4, 8, 5, 11, 5, 3, 9, 7, 5, 5, 5, 3, 8, 6, 11, 11, 2, 7, 7, 6, 5, 4, 6, 3, 8, 5, 8, 7, 6, 9, 4, 3, 7, 6, 6, 6, 5, 6, 10, 5, 9, 9, 8, 8, 7, 4, 8, 4, 9, 8, 5, 10, 10, 9, 7, 9, 7, 7, 10, 4, 7, 8, 4, 7, 8, 9, 11, 7, 9, 10, 10, 2, 7, 9, 4, 8, 8, 12, 9, 5, 11, 10, 7, 6, 4, 8, 9, 9, 6, 5, 6, 5, 6, 11, 7, 3, 10, 7, 3, 7, 7, 10, 3, 6, 8, 6, 8, 5, 10, 2, 7, 4, 8, 11, 9, 3, 4, 2, 8, 8, 6, 6, 12, 11, 10, 10, 10, 8, 4, 9, 4, 4, 6, 6, 7, 8, 2, 5, 7, 6, 9, 5, 5, 8, 4, 7, 7, 7, 6, 5, 6, 8, 6, 5, 7, 8, 4, 9, 8, 8, 9, 7, 2, 8, 3, 5, 5, 10, 7, 9, 12, 6, 4, 5, 7, 6, 4, 7, 6, 10, 3, 8, 5, 7, 7, 3, 6, 7, 7, 6, 6, 9, 12, 9, 10, 7, 10, 8, 10, 3, 9, 9, 4, 7, 8, 6, 8, 12, 5, 6, 2, 4, 4, 5, 5, 8, 7, 9, 10, 6, 7, 10, 7, 6, 8, 9, 8, 10, 3, 7, 8, 8, 8, 4, 7, 7, 8, 3, 8, 5, 9, 2, 8, 6, 11, 7, 8, 7, 6, 8, 5, 5, 3, 6, 7, 9, 7, 11, 5, 8, 2, 11, 9, 9, 7, 12, 8, 6, 9, 7, 7, 5, 7, 6, 9, 2, 5, 4, 11, 10, 4, 7, 11, 9, 8, 3, 9, 5, 5, 2, 6, 7, 8, 10, 5, 9, 4, 4, 4, 7, 6, 3, 5, 6, 4, 3, 12, 7, 7, 6, 7, 7, 4, 5, 7, 9, 3, 5, 7, 6, 11, 5, 6, 6, 8, 6, 4, 5, 7, 4, 6, 10, 3, 6, 5, 7, 6, 8, 10, 7, 7, 9, 3, 10, 9, 6, 8, 5, 9, 6, 6, 5, 3, 4, 10, 8, 10, 6, 8, 4, 7, 7, 7, 12, 5, 5, 7, 9, 5, 5, 7, 8, 7, 10, 5, 10, 7, 7, 7, 4, 7, 4, 9, 11, 8, 7, 6, 9, 7, 2, 4, 8, 4, 7, 3, 7, 8, 7, 11, 7, 5, 11, 10, 6, 7, 11, 4, 7, 5, 9, 6, 11, 6, 10, 10, 6, 10, 8, 10, 5, 5, 10, 6, 3, 7, 5, 6, 7, 12, 10, 4, 2, 6, 9, 9, 8, 7, 5, 5, 10, 7, 5, 10, 10, 6, 8, 8, 4, 6, 7, 8, 10, 4, 6, 11, 5, 3, 7, 5, 7, 12, 4, 8, 6, 11, 6, 8, 7, 11, 12, 11, 9, 8, 9, 6, 6, 2, 3, 4, 9, 3, 5, 8, 9, 7, 6, 8, 6, 4, 8, 7, 4, 6, 5, 3, 8, 4, 9, 12, 6, 9, 7, 9, 3, 8, 2, 8, 10, 5, 8, 5, 2, 3, 6, 6, 6, 9, 5, 4, 6, 5, 6, 5, 8, 10, 8, 7, 10, 9, 9, 5, 2, 11, 3, 10, 4, 9, 7, 11, 9, 2, 10, 4, 11, 5, 12, 11, 9, 8, 3, 8, 9, 5, 10, 8, 4, 6, 8, 8, 7, 6, 4, 6, 4, 9, 8, 8, 4, 7, 10, 8, 5, 6, 10, 7, 7, 3, 11, 8, 10, 9, 9, 4, 7, 10, 5, 7, 3, 4, 12, 5, 10, 12, 8, 6, 8, 4, 8, 6, 12, 5, 5, 7, 3, 2, 8, 4, 4, 10, 4, 8, 5, 5, 9, 12, 8, 5, 8, 11, 12, 5, 6, 8, 3, 7, 11, 11, 8, 4, 4, 3, 8, 11, 8, 5, 6, 11, 10, 7, 6, 6, 12, 5, 7, 5, 6, 8, 6, 8, 8, 7, 7, 5, 8, 8, 5, 10, 8, 7, 8, 7, 7, 5, 4, 7, 9, 11, 9, 8, 10, 7, 5, 6, 5, 11, 11, 9, 4, 9, 6, 6, 5, 6, 9, 7, 9, 6, 3, 6, 6, 4, 10, 12, 9, 7, 6, 7, 7, 3, 12, 7, 6, 11, 9, 7, 6, 5, 9, 8, 10, 6, 8, 6, 6, 8, 11, 7, 5, 9, 11, 3, 8, 8, 8, 9, 10, 7, 8, 7, 5, 5, 9, 8, 10, 11, 3, 6, 5, 10, 6, 7, 8, 4, 5, 4, 6, 8, 12, 10, 7, 9, 8, 7, 9, 4, 8, 7, 2, 8, 4, 3, 7, 11, 5, 4, 7, 8, 7, 8, 7, 7, 6, 4, 6, 12, 12, 7, 6, 11, 12, 10, 5, 6, 6, 5, 5, 9, 9, 8, 6, 9, 3, 11, 8, 6, 4, 8, 9, 6, 7, 9, 8, 9, 6, 6, 4, 2, 9, 8, 7, 9, 9, 3, 4, 8, 3, 8, 3, 10, 7, 7, 6, 8, 4, 5, 7, 10, 9, 8, 8, 3, 7, 6, 4, 2, 4, 3, 9, 12, 4, 9, 12, 8, 2, 9, 4, 11, 2, 6, 5, 3, 11, 4, 5, 5, 11, 4, 4, 12, 7, 4, 10, 10, 3, 8, 6, 4, 5, 6, 7, 7, 9, 3, 6, 10, 8, 3, 4, 6, 2, 6, 7, 8, 5, 8, 11, 9, 4, 8, 7, 5, 6, 11, 4, 8, 6, 6, 6, 10, 9, 4, 8, 8, 6, 9, 5, 6, 7, 7, 6, 8, 5, 10, 8, 7, 9, 6, 5, 8, 7, 8, 4, 5, 9, 10, 6, 5, 8, 8, 7, 7, 8, 5, 10, 7, 8, 5, 3, 8, 3, 9, 8, 7, 12, 7, 9, 11, 4, 4, 4, 6, 6, 8, 9, 9, 7, 7, 5, 11, 8, 8, 10, 11, 5, 7, 8, 8, 8, 7, 4, 6, 11, 11, 7, 12, 7, 11, 6, 9, 11, 6, 2, 2, 9, 7, 7, 7, 11, 9, 6, 7, 6, 7, 7, 7, 9, 8, 10, 10, 4, 7, 6, 9, 5, 7, 9, 5, 9, 5, 6, 8, 9, 10, 7, 4, 8, 5, 7, 5, 5, 9, 9, 4, 6, 4, 5, 3, 7, 8, 10, 3, 5, 11, 9, 12, 5, 8, 4, 7, 4, 7, 5, 5, 8, 4, 4, 9, 8, 7, 4, 5, 10, 9, 7, 7, 5, 8, 6, 12, 9, 7, 6, 10, 4, 7, 5, 5, 8, 5, 6, 7, 8, 9, 9, 7, 10, 8, 6, 8, 7, 5, 7, 6, 6, 5, 4, 8, 5, 3, 9, 3, 2, 9, 3, 6, 7, 7, 6, 6, 9, 6, 4, 6, 8, 5, 6, 6, 4, 8, 9, 9, 10, 7, 6, 4, 7, 4, 8, 11, 6, 7, 9, 6, 10, 6, 7, 5, 9, 6, 9, 5, 7, 5, 10, 5, 6, 7, 5, 10, 6, 8, 6, 5, 8, 4, 7, 9, 9, 7, 7, 5, 11, 7, 3, 4, 8, 6, 10, 6, 6, 6, 4, 11, 8, 5, 10, 11, 9, 8, 11, 10, 7, 9, 3, 9, 8, 8, 2, 3, 6, 11, 8, 7, 9, 12, 9, 7, 8, 5, 8, 10, 8, 9, 5, 4, 3, 8, 11, 5, 7, 5, 3, 9, 9, 7, 7, 8, 11, 7, 9, 6, 6, 10, 3, 9, 6, 7, 3, 6, 10, 5, 7, 6, 7, 6, 10, 8, 7, 4, 3, 9, 10, 5, 9, 7, 10, 2, 9, 8, 3, 7, 6, 7, 6, 5, 6, 8, 8, 11, 6, 8, 6, 6, 7, 6, 5, 6, 9, 4, 6, 6, 7, 5, 7, 4, 11, 4, 12, 6, 7, 12, 7, 5, 9, 7, 6, 5, 8, 7, 7, 6, 9, 2, 6, 4, 8, 5, 10, 7, 5, 10, 5, 6, 7, 5, 8, 10, 9, 2, 7, 4, 9, 8, 9, 7, 8, 7, 8, 5, 7, 7, 6, 9, 10, 5, 5, 6, 8, 8, 3, 4, 3, 6, 4, 9, 6, 6, 6, 8, 10, 3, 11, 5, 4, 5, 4, 8, 6, 4, 10, 8, 7, 7, 8, 8, 8, 12, 7, 6, 5, 6, 9, 6, 10, 5, 3, 6, 2, 8, 10, 10, 7, 9, 7, 7, 7, 8, 3, 6, 7, 6, 10, 8, 11, 8, 6, 5, 9, 6, 11, 9, 3, 10, 7, 7, 7, 6, 7, 9, 9, 9, 9, 10, 8, 11, 4, 6, 8, 7, 7, 10, 11, 7, 7, 7, 11, 7, 5, 10, 2, 7, 4, 3, 7, 8, 8, 4, 12, 2, 7, 5, 8, 9, 4, 9, 7, 3, 6, 8, 9, 6, 5, 4, 10, 7, 10, 9, 8, 2, 9, 7, 12, 7, 5, 6, 12, 7, 7, 5, 9, 9, 8, 8, 9, 5, 3, 8, 11, 9, 8, 8, 9, 8, 7, 6, 8, 7, 12, 7, 7, 4, 9, 10, 2, 6, 3, 5, 8, 4, 8, 7, 10, 11, 3, 6, 2, 9, 7, 12, 7, 9, 7, 9, 7, 8, 7, 3, 9, 6, 5, 10, 7, 6, 2, 9, 4, 7, 4, 7, 5, 8, 2, 5, 7, 9, 8, 10, 2, 7, 8, 9, 9, 6, 4, 10, 8, 3, 4, 5, 11, 7, 6, 8, 6, 12, 8, 8, 12, 9, 8, 7, 7, 7, 8, 3, 6, 5, 7, 7, 6, 10, 2, 8, 4, 5, 7, 8, 7, 8, 10, 12, 9, 8, 7, 9, 8, 10, 9, 9, 3, 10, 10, 8, 9, 8, 7, 11, 7, 8, 5, 7, 4, 5, 10, 7, 7, 8, 8, 5, 5, 7, 11, 11, 5, 6, 8, 8, 7, 7, 9, 5, 7, 7, 7, 9, 2, 4, 7, 9, 4, 8, 10, 7, 12, 12, 5, 2, 5, 8, 11, 7, 9, 7, 8, 7, 7, 6, 7, 10, 9, 6, 7, 2, 7, 4, 9, 8, 10, 7, 6, 4, 7, 4, 5, 7, 8, 8, 3, 8, 6, 7, 4, 11, 10, 8, 4, 12, 11, 8, 8, 3, 8, 11, 5, 4, 9, 6, 5, 8, 7, 10, 8, 12, 9, 8, 4, 7, 8, 8, 5, 9, 3, 5, 11, 6, 8, 6, 7, 11, 4, 7, 7, 7, 6, 7, 12, 9, 8, 11, 7, 7, 8, 9, 9, 6, 6, 2, 9, 6, 10, 12, 3, 4, 3, 9, 5, 7, 7, 6, 3, 6, 7, 7, 10, 11, 9, 10, 9, 10, 11, 4, 5, 12, 3, 6, 11, 3, 4, 8, 6, 9, 7, 9, 12, 9, 7, 6, 7, 10, 7, 4, 6, 5, 5, 11, 9, 8, 6, 6, 12, 4, 4, 8, 10, 12, 10, 5, 6, 4, 12, 7, 7, 7, 7, 7, 7, 8, 3, 6, 9, 3, 7, 6, 9, 7, 5, 7, 2, 12, 6, 7, 11, 9, 9, 4, 7, 11, 11, 6, 12, 3, 4, 11, 8, 8, 10, 3, 4, 4, 9, 8, 9, 7, 3, 5, 6, 9, 8, 4, 8, 3, 11, 4, 7, 7, 4, 9, 10, 8, 9, 11, 9, 5, 5, 12, 5, 9, 4, 5, 5, 10, 5, 9, 10, 8, 10, 11, 4, 4, 5, 9, 6, 7, 9, 7, 7, 8, 3, 6, 12, 7, 5, 6, 11, 5, 6, 7, 5, 10, 6, 9, 8, 7, 6, 6, 6, 2, 11, 10, 6, 10, 7, 5, 9, 7, 6, 6, 11, 9, 3, 2, 6, 6, 5, 11, 7, 7, 8, 2, 2, 9, 5, 6, 8, 2, 10, 9, 3, 7, 6, 4, 7, 8, 10, 6, 9, 2, 4, 7, 6, 9, 6, 10, 10, 8, 6, 7, 9, 7, 6, 8, 10, 10, 12, 6, 5, 4, 7, 7, 7, 11, 9, 3, 5, 11, 4, 10, 8, 7, 5, 8, 7, 3, 7, 5, 8, 7, 11, 11, 9, 5, 5, 6, 10, 9, 7, 7, 7, 10, 11, 4, 2, 7, 10, 10, 12, 5, 8, 6, 10, 9, 10, 6, 10, 10, 11, 8, 9, 6, 9, 8, 4, 8, 7, 7, 5, 6, 6, 5, 6, 2, 7, 3, 8, 7, 11, 10, 6, 7, 7, 6, 6, 3, 7, 10, 9, 6, 5, 8, 7, 2, 12, 5, 10, 8, 6, 5, 2, 4, 4, 4, 2, 6, 7, 6, 7, 7, 7, 7, 2, 7, 11, 3, 5, 5, 5, 10, 11, 7, 3, 9, 9, 3, 11, 7, 7, 7, 7, 8, 11, 10, 7, 7, 7, 7, 7, 9, 8, 4, 2, 5, 12, 8, 8, 3, 10, 10, 5, 5, 3, 9, 9, 8, 6, 6, 9, 5, 7, 11, 8, 7, 4, 9, 4, 9, 6, 5, 8, 5, 7, 9, 4, 10, 7, 5, 12, 5, 8, 7, 6, 7, 10, 5, 9, 7, 4, 6, 5, 6, 7, 7, 7, 7, 9, 10, 10, 6, 4, 5, 5, 6, 12, 12, 12, 10, 6, 8, 11, 8, 6, 8, 8, 10, 12, 8, 4, 8, 6, 9, 3, 5, 6, 8, 10, 8, 10, 8, 4, 8, 6, 9, 9, 6, 10, 10, 6, 9, 10, 7, 5, 6, 9, 2, 10, 9, 8, 9, 9, 12, 10, 7, 6, 8, 5, 5, 3, 7, 10, 6, 7, 8, 7, 7, 11, 8, 4, 8, 9, 11, 4, 7, 4, 4, 3, 5, 5, 10, 3, 10, 7, 8, 4, 5, 7, 4, 2, 11, 4, 9, 8, 3, 10, 8, 8, 5, 7, 10, 8, 6, 6, 8, 4, 6, 6, 7, 5, 6, 8, 5, 6, 6, 10, 10, 5, 7, 5, 6, 12, 2, 7, 6, 4, 8, 9, 8, 7, 6, 3, 11, 9, 2, 10, 5, 8, 8, 8, 2, 8, 5, 6, 4, 10, 3, 9, 6, 9, 9, 9, 8, 7, 4, 7, 8, 8, 12, 10, 4, 9, 7, 3, 5, 6, 3, 6, 5, 5, 4, 4, 7, 8, 10, 8, 10, 11, 6, 6, 11, 5, 7, 11, 4, 6, 2, 9, 10, 10, 11, 10, 7, 8, 5, 6, 11, 7, 10, 3, 6, 3, 2, 6, 8, 7, 3, 8, 7, 11, 11, 7, 6, 7, 11, 7, 5, 7, 6, 3, 11, 5, 6, 7, 9, 7, 8, 7, 8, 10, 7, 9, 3, 10, 7, 10, 12, 4, 6, 7, 3, 8, 7, 7, 10, 5, 6, 3, 6, 7, 7, 12, 6, 4, 7, 9, 4, 5, 6, 8, 7, 7, 10, 9, 4, 6, 7, 8, 4, 9, 9, 5, 10, 8, 2, 5, 3, 7, 5, 9, 4, 5, 9, 5, 10, 4, 5, 6, 7, 6, 8, 8, 10, 9, 10, 7, 7, 7, 5, 5, 3, 6, 8, 6, 5, 8, 6, 4, 7, 3, 9, 9, 8, 7, 8, 8, 7, 2, 3, 6, 12, 3, 7, 6, 8, 9, 7, 7, 6, 12, 5, 10, 10, 9, 7, 6, 4, 12, 5, 4, 10, 5, 9, 7, 5, 7, 6, 5, 7, 7, 6, 5, 10, 11, 6, 4, 10, 6, 7, 7, 5, 2, 7, 7, 6, 8, 8, 11, 6, 5, 6, 8, 9, 7, 8, 6, 7, 4, 6, 4, 7, 4, 7, 11, 5, 6, 9, 7, 3, 11, 9, 7, 7, 2, 6, 9, 8, 8, 6, 2, 8, 5, 5, 9, 10, 7, 8, 7, 7, 8, 11, 3, 7, 8, 7, 3, 3, 6, 9, 5, 7, 10, 4, 5, 9, 6, 6, 6, 11, 9, 9, 7, 10, 4, 9, 6, 11, 6, 3, 5, 4, 3, 11, 10, 7, 7, 4, 3, 7, 10, 11, 7, 3, 8, 5, 7, 9, 7, 4, 9, 4, 5, 5, 4, 8, 5, 8, 8, 7, 6, 7, 9, 7, 9, 7, 9, 2, 9, 11, 5, 3, 9, 9, 6, 5, 10, 8, 5, 5, 7, 4, 6, 9, 7, 5, 8, 8, 7, 5, 5, 7, 9, 3, 5, 5, 7, 10, 8, 6, 6, 10, 9, 5, 4, 12, 2, 10, 5, 7, 7, 6, 10, 5, 8, 4, 3, 8, 7, 7, 11, 7, 10, 10, 10, 6, 6, 3, 10, 8, 9, 7, 6, 12, 7, 6, 7, 5, 9, 6, 5, 9, 9, 5, 9, 9, 6, 5, 11, 8, 9, 6, 4, 5, 3, 3, 11, 5, 9, 7, 3, 10, 9, 12, 9, 2, 7, 10, 4, 10, 3, 3, 11, 2, 4, 8, 6, 6, 8, 12, 8, 12, 9, 6, 4, 8, 3, 3, 6, 6, 11, 9, 10, 8, 7, 11, 11, 9, 7, 7, 8, 9, 9, 6, 9, 7, 7, 11, 9, 4, 11, 9, 8, 8, 6, 6, 7, 10, 9, 2, 9, 6, 7, 11, 7, 2, 9, 5, 11, 5, 8, 8, 6, 8, 9, 8, 6, 6, 10, 4, 9, 3, 8, 6, 8, 12, 8, 7, 8, 7, 2, 8, 6, 6, 6, 5, 9, 5, 5, 10, 6, 8, 9, 4, 5, 7, 8, 9, 7, 5, 4, 5, 7, 5, 8, 11, 4, 7, 5, 5, 8, 5, 12, 9, 4, 3, 2, 7, 11, 8, 4, 9, 6, 5, 7, 7, 9, 7, 10, 4, 7, 12, 6, 3, 11, 5, 8, 9, 9, 6, 7, 3, 9, 9, 9, 5, 5, 7, 3, 8, 7, 2, 7, 7, 3, 7, 6, 5, 6, 8, 3, 6, 6, 4, 11, 9, 4, 6, 8, 5, 7, 7, 7, 4, 10, 11, 6, 7, 6, 7, 6, 8, 4, 5, 5, 7, 4, 6, 9, 6, 4, 7, 9, 10, 11, 3, 5, 5, 9, 2, 7, 11, 6, 7, 7, 9, 8, 9, 8, 4, 4, 4, 10, 6, 9, 8, 5, 5, 7, 8, 5, 6, 10, 5, 5, 4, 9, 8, 3, 9, 5, 12, 10, 4, 9, 12, 7, 10, 11, 3, 11, 10, 7, 5, 7, 7, 11, 6, 8, 6, 5, 12, 6, 8, 10, 4, 5, 4, 5, 5, 7, 7, 2, 5, 8, 5, 4, 8, 5, 6, 11, 10, 9, 6, 6, 4, 10, 9, 9, 7, 5, 10, 6, 8, 9, 3, 3, 7, 4, 8, 10, 9, 7, 6, 7, 5, 11, 6, 2, 7, 5, 9, 12, 5, 5, 2, 8, 5, 12, 4, 8, 4, 6, 7, 8, 6, 8, 9, 8, 6, 7, 6, 11, 4, 10, 2, 4, 4, 8, 3, 8, 2, 8, 5, 4, 4, 7, 6, 8, 10, 6, 4, 6, 5, 9, 5, 6, 10, 7, 6, 8, 7, 9, 7, 7, 4, 11, 9, 3, 8, 11, 4, 7, 6, 5, 8, 4, 5, 7, 11, 6, 7, 5, 8, 7, 7, 9, 5, 9, 4, 2, 9, 12, 6, 12, 7, 11, 6, 6, 7, 9, 4, 3, 3, 7, 6, 8, 6, 6, 9, 3, 6, 12, 5, 4, 3, 10, 8, 8, 7, 6, 12, 6, 10, 11, 3, 7, 9, 10, 11, 7, 11, 3, 8, 7, 9, 5, 9, 6, 7, 9, 3, 6, 3, 6, 7, 5, 8, 5, 8, 5, 9, 10, 3, 7, 8, 4, 8, 3, 7, 5, 7, 10, 5, 7, 11, 6, 2, 7, 7, 8, 7, 7, 7, 8, 4, 7, 4, 3, 7, 7, 9, 6, 8, 11, 3, 8, 6, 7, 4, 7, 10, 7, 7, 5, 8, 4, 7, 10, 6, 8, 12, 9, 6, 8, 6, 9, 4, 11, 7, 5, 5, 6, 6, 7, 7, 6, 12, 12, 5, 10, 6, 3, 6, 9, 7, 11, 6, 8, 5, 7, 9, 7, 3, 5, 6, 8, 9, 9, 8, 7, 6, 10, 7, 11, 8, 7, 7, 10, 6, 8, 9, 4, 7, 5, 7, 9, 8, 7, 11, 12, 3, 10, 12, 9, 7, 6, 8, 8, 5, 6, 9, 9, 6, 3, 8, 6, 4, 5, 9, 7, 8, 7, 7, 5, 9, 8, 8, 4, 8, 5, 7, 5, 11, 7, 10, 5, 4, 5, 6, 3, 3, 8, 10, 3, 8, 9, 7, 7, 7, 5, 11, 9, 7, 7, 7, 6, 4, 10, 5, 9, 2, 7, 9, 6, 6, 11, 10, 9, 4, 6, 5, 10, 9, 8, 6, 8, 9, 8, 10, 5, 5, 8, 9, 9, 7, 5, 7, 10, 11, 7, 6, 9, 6, 7, 10, 8, 7, 4, 11, 7, 8, 8, 7, 8, 6, 11, 12, 9, 9, 5, 7, 7, 7, 7, 2, 6, 7, 5, 9, 3, 4, 6, 6, 2, 5, 6, 4, 8, 11, 2, 8, 5, 7, 7, 6, 7, 6, 4, 7, 6, 3, 9, 9, 11, 7, 11, 7, 10, 8, 4, 5, 2, 7, 9, 8, 3, 7, 3, 5, 6, 8, 12, 11, 5, 2, 2, 5, 7, 12, 7, 6, 5, 5, 5, 8, 11, 12, 3, 6, 6, 6, 2, 7, 6, 3, 7, 4, 5, 8, 7, 4, 11, 7, 4, 11, 5, 9, 6, 10, 6, 6, 6, 8, 6, 12, 10, 5, 12, 4, 8, 5, 5, 8, 11, 9, 12, 4, 5, 4, 3, 7, 7, 6, 10, 4, 4, 7, 7, 6, 7, 8, 10, 8, 6, 9, 9, 4, 7, 8, 6, 8, 7, 6, 6, 7, 10, 11, 7, 5, 5, 3, 4, 11, 5, 4, 9, 9, 4, 6, 5, 5, 12, 9, 8, 10, 9, 12, 4, 4, 5, 10, 3, 6, 7, 5, 6, 9, 8, 9, 6, 6, 4, 7, 6, 6, 6, 6, 8, 7, 10, 8, 10, 12, 3, 11, 12, 7, 8, 7, 6, 2, 12, 7, 6, 10, 6, 8, 8, 8, 5, 10, 8, 9, 7, 7, 3, 8, 6, 10, 9, 4, 7, 7, 7, 9, 3, 7, 3, 4, 8, 7, 6, 12, 12, 12, 7, 2, 8, 4, 5, 3, 4, 11, 10, 6, 4, 11, 5, 6, 11, 2, 7, 9, 7, 7, 12, 6, 4, 7, 9, 7, 8, 7, 6, 7, 10, 5, 7, 9, 4, 7, 12, 5, 7, 8, 9, 9, 9, 10, 4, 10, 10, 6, 9, 9, 4, 6, 9, 9, 8, 7, 8, 7, 10, 8, 6, 5, 6, 6, 4, 10, 8, 10, 6, 11, 9, 7, 8, 11, 8, 5, 4, 10, 4, 6, 7, 9, 3, 6, 7, 7, 6, 10, 8, 7, 7, 5, 2, 4, 7, 7, 7, 8, 8, 10, 7, 7, 4, 7, 9, 10, 9, 6, 7, 8, 3, 4, 7, 8, 11, 9, 9, 9, 7, 11, 11, 10, 3, 9, 5, 10, 10, 7, 2, 7, 8, 8, 8, 8, 12, 10, 8, 9, 4, 7, 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7, 6, 7, 11, 11, 10, 12, 8, 5, 5, 3, 7, 6, 9, 6, 12, 12, 10, 7, 9, 4, 8, 6, 8, 9, 9, 10, 4, 5, 9, 3, 6, 9, 7, 8, 8, 6, 10, 10, 5, 11, 9, 4, 4, 9, 10, 9, 7, 4, 7, 8, 5, 7, 4, 4, 6, 8, 5, 8, 8, 5, 10, 7, 11, 8, 2, 5, 7, 8, 10, 6, 9, 6, 7, 4, 9, 3, 5, 9, 6, 3, 11, 12, 10, 3, 8, 5, 10, 8, 5, 4, 8, 9, 8, 11, 6, 9, 9, 6, 11, 5, 4, 9, 10, 8, 3, 8, 6, 8, 11, 10, 5, 5, 7, 9, 3, 11, 6, 9, 8, 9, 7, 2, 8, 4, 7, 5, 10, 9, 7, 5, 9, 5, 9, 5, 8, 5, 6, 5, 8, 8, 7, 7, 3, 6, 3, 8, 7, 11, 6, 3, 11, 11, 9, 8, 6, 4, 9, 6, 10, 5, 4, 4, 3, 6, 6, 5, 8, 8, 2, 10, 6, 8, 7, 9, 4, 8, 8, 8, 5, 5, 2, 8, 9, 10, 8, 7, 5, 5, 5, 12, 4, 7, 8, 11, 9, 8, 11, 11, 10, 9, 5, 7, 10, 4, 6, 5, 10, 8, 12, 7, 6, 3, 7, 5, 8, 9, 6, 9, 11, 9, 11, 12, 6, 5, 8, 9, 8, 9, 4, 5, 4, 2, 6, 3, 10, 3, 7, 8, 10, 7, 11, 10, 8, 7, 6, 9, 5, 6, 5, 11, 7, 9, 2, 7, 6, 3, 6, 3, 8, 5, 3, 4, 2, 11, 4, 9, 9, 7, 5, 6, 7, 7, 8, 7, 9, 4, 9, 6, 4, 8, 10, 9, 8, 3, 4, 8, 5, 9, 5, 7, 11, 4, 6, 2, 9, 4, 9, 10, 8, 7, 8, 9, 5, 6, 9, 6, 10, 8, 8, 4, 6, 4, 4, 5, 9, 7, 4, 9, 7, 4, 7, 9, 6, 2, 5, 10, 5, 6, 3, 11, 9, 10, 8, 5, 6, 7, 2, 6, 9, 6, 7, 8, 6, 6, 6, 6, 10, 9, 8, 9, 7, 9, 7, 6, 9, 7, 3, 9, 8, 10, 4, 9, 4, 8, 3, 8, 7, 10, 8, 5, 10, 4, 10, 5, 6, 6, 7, 7, 8, 9, 7, 6, 3, 8, 3, 10, 10, 6, 11, 6, 11, 9, 9, 8, 7, 7, 10, 4, 9, 5, 12, 12, 4, 10, 11, 7, 8, 7, 3, 9, 9, 5, 6, 6, 3, 7, 4, 10, 7, 6, 4, 7, 7, 3, 4, 5, 4, 10, 6, 5, 7, 10, 12, 7, 12, 8, 8, 6, 10, 3, 5, 12, 7, 5, 5, 10, 7, 5, 11, 6, 5, 4, 4, 5, 11, 6, 2, 12, 6, 7, 6, 8, 7, 6, 11, 8, 8, 9, 6, 5, 7, 2, 9, 9, 7, 9, 7, 3, 8, 7, 7, 5, 4, 6, 7, 8, 8, 8, 4, 3, 6, 7, 11, 7, 5, 4, 6, 11, 8, 5, 4, 2, 6, 10, 12, 3, 6, 6, 4, 4, 9, 5, 9, 9, 5, 4, 5, 8, 7, 4, 10, 5, 6, 7, 10, 9, 11, 11, 9, 10, 10, 11, 8, 6, 2, 11, 9, 3, 5, 6, 6, 7, 11, 7, 3, 9, 12, 8, 5, 6, 6, 5, 10, 9, 7, 4, 7, 6, 7, 9, 5, 9, 9, 7, 7, 2, 11, 10, 10, 4, 12, 4, 5, 12, 10, 10, 6, 11, 5, 7, 6, 4, 8, 7, 8, 7, 10, 2, 8, 6, 4, 5, 12, 4, 6, 5, 6, 11, 2, 10, 7, 11, 5, 5, 9, 8, 8, 8, 9, 2, 11, 6, 10, 8, 7, 2, 8, 9, 8, 9, 4, 12, 7, 8, 5, 4, 9, 6, 3, 5, 8, 9, 11, 9, 11, 4, 5, 4, 7, 9, 6, 10, 8, 11, 3, 5, 4, 6, 4, 8, 10, 8, 11, 3, 11, 6, 4, 10, 6, 5, 4, 4, 8, 6, 8, 6, 8, 6, 5, 2, 8, 4, 8, 3, 5, 9, 7, 10, 4, 5, 10, 4, 11, 4, 7, 9, 7, 11, 4, 8, 3, 7, 9, 2, 10, 3, 7, 10, 3, 11, 4, 7, 6, 8, 2, 6, 3, 8, 2, 10, 5, 6, 6, 8, 5, 7, 7, 7, 5, 5, 11, 10, 6, 2, 5, 8, 4, 7, 6, 7, 8, 7, 9, 12, 9, 7, 2, 8, 8, 9, 8, 8, 7, 8, 4, 9, 5, 7, 7, 11, 6, 10, 6, 4, 10, 6, 5, 7, 6, 7, 6, 10, 9, 8, 5, 6, 8, 8, 8, 6, 5, 7, 5, 7, 9, 4, 10, 6, 8, 7, 9, 5, 8, 6, 5, 8, 6, 5, 9, 6, 3, 4, 10, 8, 10, 9, 6, 7, 5, 11, 11, 8, 3, 6, 6, 5, 12, 6, 8, 3, 3, 4, 7, 7, 5, 7, 8, 6, 10, 9, 3, 6, 8, 4, 6, 11, 6, 6, 11, 7, 9, 8, 8, 8, 8, 4, 6, 10, 5, 7, 9, 6, 10, 4, 11, 7, 4, 10, 6, 3, 10, 4, 7, 5, 7, 4, 9, 9, 7, 8, 6, 8, 10, 5, 8, 2, 6, 3, 3, 8, 12, 6, 9, 10, 11, 10, 6, 5, 7, 10, 8, 6, 10, 7, 8, 7, 6, 8, 12, 5, 7, 5, 8, 3, 7, 7, 12, 10, 12, 10, 11, 6, 3, 5, 8, 3, 4, 6, 11, 8, 4, 9, 6, 9, 7, 3, 8, 7, 5, 6, 12, 4, 12, 7, 6, 6, 5, 6, 9, 10, 7, 8, 3, 5, 3, 6, 8, 7, 7, 5, 2, 7, 11, 7, 6, 9, 8, 6, 7, 8, 4, 8, 8, 5, 9, 9, 10, 6, 7, 11, 9, 9, 5, 12, 2, 5, 7, 7, 3, 9, 7, 8, 6, 8, 9, 4, 5, 6, 5, 11, 11, 6, 6, 10, 9, 4, 4, 7, 9, 11, 6, 4, 8, 9, 4, 8, 12, 4, 6, 9, 2, 4, 7, 3, 4, 5, 8, 7, 5, 7, 6, 3, 6, 10, 9, 12, 10, 9, 9, 11, 6, 10, 4, 6, 6, 2, 9, 7, 8, 5, 8, 5, 6, 10, 5, 4, 12, 2, 7, 4, 7, 6, 5, 6, 4, 8, 8, 9, 7, 9, 10, 5, 2, 6, 8, 5, 8, 8, 9, 10, 5, 5, 10, 4, 3, 5, 10, 5, 5, 4, 7, 10, 11, 2, 6, 3, 7, 4, 8, 2, 8, 6, 8, 10, 9, 8, 7, 7, 7, 5, 5, 8, 10, 8, 3, 7, 6, 6, 5, 3, 7, 5, 9, 6, 11, 6, 3, 5, 9, 3, 7, 5, 10, 9, 9, 9, 6, 3, 7, 5, 9, 7, 7, 5, 7, 7, 5, 7, 7, 12, 6, 2, 5, 10, 10, 11, 12, 7, 4, 9, 8, 6, 5, 7, 7, 8, 7, 6, 4, 4, 9, 9, 9, 10, 8, 9, 7, 5, 7, 7, 4, 6, 9, 8, 7, 7, 6, 3, 7, 7, 6, 3, 10, 9, 6, 6, 3, 11, 9, 8, 9, 12, 12, 3, 6, 10, 4, 7, 6, 4, 6, 10, 7, 10, 7, 12, 6, 9, 5, 8, 7, 5, 4, 12, 7, 7, 8, 10, 8, 7, 3, 11, 9, 9, 7, 7, 10, 6, 3, 2, 8, 8, 6, 7, 7, 10, 8, 3, 7, 6, 5, 5, 4, 10, 9, 6, 8, 3, 3, 6, 11, 10, 8, 7, 3, 9, 6, 8, 9, 7, 8, 9, 12, 10, 8, 8, 7, 6, 3, 4, 9, 6, 2, 8, 6, 10, 10, 7, 4, 7, 2, 8, 7, 4, 6, 9, 12, 12, 4, 8, 12, 9, 10, 8, 4, 6, 8, 10, 5, 11, 5, 6, 5, 8, 11, 3, 3, 6, 12, 5, 10, 5, 5, 5, 8, 8, 6, 5, 8, 10, 11, 8, 9, 7, 10, 9, 8, 10, 9, 12, 8, 9, 5, 11, 6, 6, 10, 5, 7, 11, 8, 7, 7, 9, 5, 9, 4, 5, 9, 11, 9, 4, 5, 11, 7, 6, 7, 6, 6, 8, 7, 9, 6, 7, 5, 9, 2, 4, 8, 9, 12, 5, 3, 6, 7, 8, 7, 3, 12, 7, 6, 10, 3, 5, 6, 9, 8, 2, 6, 10, 9, 12, 5, 5, 9, 4, 10, 4, 8, 7, 12, 8, 5, 9, 6, 7, 3, 6, 7, 7, 4, 5, 8, 8, 7, 5, 4, 8, 6, 5, 9, 7, 11, 9, 7, 7, 8, 11, 7, 7, 10, 7, 6, 6, 11, 6, 7, 11, 10, 9, 8, 7, 7, 10, 8, 9, 9, 12, 7, 6, 7, 4, 9, 7, 8, 8, 7, 5, 8, 11, 5, 8, 6, 8, 11, 4, 8, 9, 6, 10, 7, 6, 7, 11, 3, 9, 4, 5, 6, 9, 7, 7, 8, 6, 4, 10, 6, 5, 10, 10, 11, 8, 9, 6, 10, 11, 10, 10, 10, 6, 7, 3, 6, 9, 7, 8, 5, 7, 10, 3, 8, 9, 8, 5, 7, 9, 5, 6, 7, 7, 10, 10, 7, 6, 8, 10, 3, 7, 11, 8, 5, 9, 8, 9, 8, 4, 8, 7, 5, 9, 10, 8, 10, 7, 9, 4, 4, 12, 9, 11, 2, 6, 6, 5, 7, 5, 6, 5, 6, 7, 11, 3, 9, 9, 3, 6, 8, 6, 8, 10, 6, 8, 7, 2, 9, 3, 5, 7, 7, 5, 9, 5, 8, 5, 7, 7, 7, 8, 5, 8, 8, 6, 10, 9, 4, 6, 4, 12, 6, 7, 6, 7, 7, 9, 9, 7, 9, 4, 8, 3, 10, 10, 5, 10, 5, 7, 9, 11, 8, 7, 6, 12, 11, 8, 6, 5, 9, 3, 9, 8, 9, 7, 8, 7, 9, 8, 6, 3, 7, 8, 4, 3, 7, 6, 11, 7, 7, 9, 8, 9, 10, 3, 6, 9, 5, 8, 8, 8, 9, 8, 5, 5, 5, 7, 11, 5, 9, 9, 6, 11, 7, 11, 9, 10, 7, 6, 7, 8, 10, 4, 3, 8, 5, 7, 7, 7, 7, 9, 10, 6, 9, 4, 11, 10, 8, 8, 5, 4, 4, 6, 7, 2, 3, 4, 7, 8, 8, 7, 11, 7, 8, 7, 7, 3, 7, 7, 4, 8, 10, 8, 4, 10, 8, 11, 5, 9, 9, 7, 7, 8, 5, 4, 7, 2, 3, 7, 5, 6, 7, 8, 10, 4, 7, 8, 8, 9, 7, 7, 7, 5, 3, 9, 5, 9, 9, 8, 4, 10, 11, 6, 7, 8, 10, 5, 7, 8, 6, 4, 2, 9, 8, 7, 6, 3, 7, 12, 9, 6, 7, 12, 11, 6, 10, 3, 7, 8, 6, 6, 5, 11, 6, 11, 7, 11, 6, 10, 5, 9, 6, 8, 9, 3, 4, 8, 2, 5, 4, 11, 5, 11, 8, 5, 8, 7, 11, 3, 6, 8, 11, 8, 10, 5, 6, 9, 4, 8, 5, 7, 7, 6, 5, 9, 3, 7, 7, 6, 7, 9, 8, 4, 7, 7, 9, 7, 9, 5, 7, 6, 7, 7, 12, 4, 8, 9, 6, 7, 5, 6, 5, 3, 8, 9, 5, 10, 4, 6, 5, 8, 11, 9, 12, 4, 10, 5, 8, 7, 6, 11, 5, 8, 5, 3, 10, 5, 9, 12, 12, 10, 8, 6, 7, 11, 3, 5, 5, 8, 2, 8, 7, 5, 6, 5, 5, 9, 8, 7, 7, 7, 8, 8, 7, 10, 5, 9, 5, 4, 7, 4, 3, 5, 8, 11, 6, 7, 4, 7, 5, 5, 7, 8, 10, 10, 4, 5, 5, 6, 11, 6, 7, 6, 6, 12, 7, 8, 6, 5, 5, 2, 9, 6, 7, 8, 11, 8, 9, 12, 6, 5, 11, 11, 10, 10, 8, 7, 8, 5, 7, 9, 2, 3, 10, 8, 8, 10, 8, 4, 11, 9, 6, 7, 10, 9, 8, 9, 7, 9, 9, 6, 9, 5, 7, 11, 3, 8, 4, 12, 11, 9, 5, 7, 10, 7, 7, 9, 3, 6, 11, 9, 7, 3, 7, 6, 8, 7, 9, 4, 9, 7, 10, 8, 8, 9, 4, 5, 3, 10, 9, 5, 10, 9, 3, 9, 9, 5, 8, 10, 6, 5, 11, 7, 6, 9, 9, 3, 6, 5, 8, 7, 5, 12, 8, 6, 7, 4, 4, 7, 3, 9, 3, 10, 7, 7, 3, 5, 7, 5, 5, 8, 6, 4, 8, 2, 6, 7, 4, 11, 7, 7, 5, 6, 5, 4, 6, 7, 3, 7, 5, 4, 7, 6, 9, 4, 6, 7, 2, 6, 6, 9, 5, 9, 8, 7, 3, 11, 10, 5, 8, 7, 6, 7, 5, 11, 8, 10, 7, 8, 7, 8, 5, 8, 7, 7, 10, 4, 9, 11, 6, 7, 11, 6, 11, 10, 10, 8, 11, 8, 8, 10, 5, 5, 6, 8, 11, 7, 11, 3, 10, 7, 9, 8, 5, 5, 8, 8, 6, 10, 3, 8, 7, 4, 6, 8, 8, 10, 7, 7, 6, 4, 3, 7, 9, 6, 3, 12, 11, 12, 7, 10, 5, 7, 7, 7, 6, 6, 2, 8, 11, 7, 8, 7, 7, 7, 11, 9, 10, 6, 10, 6, 8, 9, 8, 9, 8, 3, 4, 9, 8, 7, 10, 3, 2, 12, 3, 5, 8, 7, 4, 7, 7, 10, 7, 7, 8, 5, 8, 7, 6, 7, 5, 11, 6, 2, 6, 8, 6, 5, 7, 9, 5, 9, 11, 6, 5, 5, 12, 9, 8, 5, 8, 4, 5, 11, 6, 9, 7, 9, 6, 2, 5, 10, 5, 6, 7, 5, 11, 7, 6, 5, 7, 6, 6, 7, 7, 3, 4, 8, 8, 8, 7, 9, 6, 5, 5, 3, 7, 2, 7, 6, 7, 7, 5, 7, 7, 4, 7, 9, 5, 7, 6, 7, 8, 10, 4, 10, 5, 4, 5, 7, 6, 7, 10, 4, 3, 6, 8, 8, 8, 2, 8, 10, 6, 7, 6, 8, 5, 4, 10, 3, 3, 10, 3, 8, 8, 5, 3, 7, 7, 7, 5, 7, 5, 2, 5, 10, 5, 3, 6, 9, 7, 8, 4, 5, 10, 5, 6, 10, 6, 8, 6, 7, 5, 12, 11, 3, 8, 10, 4, 6, 3, 5, 6, 6, 7, 8, 7, 6, 10, 6, 6, 9, 5, 7, 4, 8, 3, 3, 6, 9, 12, 8, 5, 10, 8, 10, 6, 9, 3, 7, 11, 3, 5, 12, 3, 8, 9, 5, 5, 7, 10, 7, 9, 6, 5, 6, 5, 4, 7, 8, 4, 10, 6, 4, 5, 8, 9, 7, 7, 7, 4, 11, 7, 10, 4, 6, 10, 6, 11, 4, 7, 3, 7, 4, 12, 10, 8, 3, 9, 9, 7, 11, 7, 12, 7, 3, 4, 3, 10, 6, 12, 5, 8, 6, 6, 2, 10, 7, 7, 7, 4, 2, 7, 10, 8, 3, 3, 7, 7, 6, 7, 10, 11, 5, 7, 8, 12, 7, 12, 3, 6, 7, 11, 8, 9, 5, 4, 7, 8, 10, 7, 7, 11, 7, 8, 8, 4, 3, 5, 6, 8, 2, 5, 4, 5, 9, 7, 3, 8, 8, 10, 9, 9, 3, 7, 4, 8, 6, 3, 5, 10, 12, 4, 5, 12, 11, 11, 5, 12, 7, 5, 8, 6, 7, 8, 2, 9, 3, 8, 7, 4, 8, 5, 10, 12, 5, 7, 4, 9, 7, 4, 8, 5, 2, 5, 12, 5, 7, 6, 5, 4, 2, 6, 8, 6, 7, 8, 6, 10, 8, 7, 9, 6, 7, 11, 9, 11, 6, 7, 4, 10, 11, 11, 9, 8, 7, 6, 6, 5, 8, 2, 3, 4, 12, 7, 11, 7, 6, 10, 7, 9, 8, 9, 7, 7, 6, 7, 11, 5, 2, 6, 5, 8, 11, 9, 10, 10, 8, 7, 11, 7, 5, 11, 8, 6, 5, 7, 5, 7, 5, 7, 10, 2, 10, 8, 7, 12, 11, 8, 11, 10, 8, 6, 11, 10, 3, 5, 7, 8, 9, 5, 10, 7, 8, 10, 7, 7, 12, 5, 11, 10, 6, 5, 8, 5, 11, 7, 8, 4, 7, 8, 6, 6, 11, 7, 5, 9, 6, 6, 5, 6, 8, 8, 10, 7, 6, 7, 7, 6, 10, 2, 12, 3, 6, 3, 7, 3, 2, 3, 3, 8, 10, 10, 7, 10, 5, 3, 9, 11, 8, 7, 5, 7, 6, 3, 8, 10, 3, 10, 6, 7, 3, 5, 3, 9, 5, 2, 5, 5, 9, 11, 9, 7, 3, 10, 10, 5, 10, 7, 9, 10, 5, 12, 9, 9, 7, 8, 10, 10, 7, 4, 6, 9, 8, 11, 11, 7, 7, 5, 8, 7, 10, 10, 8, 9, 7, 8, 3, 11, 9, 5, 11, 6, 9, 2, 2, 8, 8, 10, 9, 7, 12, 8, 6, 12, 5, 6, 2, 9, 9, 9, 7, 5, 7, 7, 7, 9, 8, 9, 11, 4, 4, 7, 8, 2, 4, 11, 11, 9, 9, 4, 4, 6, 2, 7, 10, 6, 4, 6, 10, 8, 8, 7, 7, 7, 4, 9, 6, 9, 7, 9, 6, 9, 7, 6, 8, 7, 5, 4, 6, 3, 9, 11, 5, 5, 6, 6, 8, 8, 4, 12, 4, 7, 7, 11, 8, 7, 3, 5, 8, 8, 2, 7, 8, 3, 6, 8, 6, 3, 8, 5, 3, 8, 5, 7, 7, 8, 6, 9, 7, 7, 9, 7, 10, 6, 11, 9, 7, 3, 4, 6, 6, 7, 4, 9, 9, 11, 2, 9, 3, 5, 9, 11, 9, 7, 7, 11, 9, 7, 6, 5, 10, 2, 6, 8, 7, 12, 5, 7, 6, 5, 5, 8, 8, 7, 8, 2, 8, 3, 10, 9, 7, 4, 4, 9, 6, 8, 7, 5, 9, 9, 7, 3, 7, 4, 8, 7, 9, 7, 4, 9, 7, 3, 10, 12, 9, 9, 5, 4, 6, 4, 7, 11, 6, 6, 8, 4, 11, 12, 7, 6, 6, 8, 12, 6, 7, 11, 5, 10, 8, 3, 5, 7, 5, 6, 3, 6, 10, 6, 6, 6, 11, 8, 7, 8, 9, 5, 5, 7, 6, 4, 10, 6, 7, 5, 9, 10, 4, 6, 5, 7, 8, 3, 8, 6, 3, 4, 4, 7, 11, 5, 8, 6, 7, 8, 11, 3, 8, 9, 4, 9, 9, 9, 9, 7, 6, 5, 6, 2, 5, 7, 10, 5, 6, 7, 9, 7, 8, 9, 8, 4, 11, 7, 7, 3, 4, 8, 7, 7, 7, 3, 9, 8, 5, 6, 6, 6, 7, 7, 10, 5, 8, 5, 10, 9, 8, 9, 5, 8, 9, 8, 7, 10, 5, 7, 4, 3, 8, 6, 8, 4, 5, 10, 5, 5, 8, 7, 12, 6, 4, 5, 4, 4, 5, 4, 6, 9, 8, 5, 4, 11, 8, 7, 5, 7, 7, 5, 5, 10, 6, 8, 4, 4, 4, 6, 7, 7, 4, 3, 7, 6, 10, 5, 6, 8, 5, 6, 10, 7, 9, 6, 9, 6, 8, 8, 7, 8, 7, 6, 4, 4, 9, 4, 8, 6, 10, 5, 8, 4, 11, 6, 7, 8, 7, 4, 7, 6, 9, 8, 3, 6, 9, 5, 6, 6, 11, 8, 8, 7, 7, 2, 12, 7, 4, 2, 2, 6, 10, 9, 4, 3, 6, 11, 6, 10, 7, 4, 7, 7, 8, 11, 3, 5, 4, 9, 4, 5, 8, 7, 5, 10, 6, 6, 9, 8, 8, 12, 8, 8, 9, 9, 6, 4, 7, 6, 10, 8, 6, 8, 6, 9, 6, 8, 5, 4, 2, 5, 11, 7, 7, 7, 3, 7, 5, 4, 4, 9, 5, 7, 10, 4, 8, 5, 4, 7, 10, 10, 5, 5, 11, 6, 9, 4, 6, 8, 2, 4, 4, 9, 5, 10, 2, 12, 11, 5, 10, 7, 7, 2, 11, 6, 7, 6, 11, 5, 5, 7, 7, 7, 4, 10, 5, 11, 7, 9, 4, 4, 5, 10, 7, 9, 7, 9, 4, 8, 8, 7, 7, 8, 6, 7, 9, 8, 4, 3, 7, 2, 10, 9, 7, 11, 6, 8, 11, 8, 4, 11, 9, 3, 10, 8, 7, 11, 5, 7, 4, 2, 9, 6, 3, 12, 4, 11, 9, 8, 6, 9, 8, 11, 10, 7, 7, 3, 4, 7, 6, 7, 5, 6, 6, 8, 4, 6, 9, 5, 3, 6, 3, 2, 2, 3, 10, 5, 9, 4, 4, 9, 5, 4, 7, 9, 7, 12, 3, 8, 4, 8, 8, 9, 5, 9, 8, 11, 6, 6, 8, 10, 6, 8, 8, 4, 8, 6, 7, 7, 8, 6, 8, 2, 11, 3, 7, 3, 6, 11, 7, 5, 9, 10, 10, 3, 6, 8, 4, 10, 9, 8, 8, 10, 10, 4, 10, 8, 3, 9, 10, 7, 9, 7, 2, 5, 9, 8, 9, 10, 9, 9, 11, 6, 6, 7, 10, 3, 7, 6, 8, 10, 5, 6, 3, 11, 6, 4, 7, 5, 4, 5, 8, 9, 8, 7, 8, 3, 6, 3, 7, 10, 10, 5, 9, 8, 9, 5, 10, 10, 12, 7, 4, 4, 7, 9, 9, 11, 6, 7, 7, 5, 7, 8, 5, 9, 11, 9, 9, 8, 9, 9, 3, 8, 8, 7, 10, 9, 3, 8, 7, 5, 5, 6, 6, 8, 8, 7, 9, 8, 2, 4, 3, 12, 9, 4, 5, 5, 5, 8, 8, 5, 2, 7, 6, 5, 9, 5, 9, 9, 3, 7, 9, 8, 12, 9, 4, 8, 11, 6, 4, 11, 7, 9, 4, 6, 6, 11, 9, 5, 9, 4, 7, 10, 9, 6, 6, 5, 9, 6, 6, 3, 4, 7, 8, 8, 4, 9, 8, 8, 7, 10, 8, 9, 6, 11, 6, 6, 8, 9, 10, 6, 4, 4, 5, 6, 7, 9, 12, 6, 7, 8, 7, 7, 10, 9, 4, 7, 10, 9, 5, 2, 8, 6, 7, 5, 7, 8, 7, 8, 8, 6, 7, 9, 7, 7, 3, 7, 6, 7, 8, 12, 10, 7, 9, 6, 5, 8, 7, 7, 10, 6, 4, 10, 3, 8, 9, 6, 6, 8, 3, 4, 9, 5, 3, 11, 10, 12, 5, 5, 10, 6, 6, 10, 10, 4, 6, 11, 9, 10]
Note that this data, on its own, represents an approximation for the probability mass function. If you wanted to approximate

A normalized histogram (where each value is divided by the length of your data list) is an approximation of the PMF. For a dataset of discrete numbers, a histogram shows the count of each value (in this case
In this case, since we ran 10,000 trials, the histogram is a very good approximation of the PMF. We use the sum of dice as an example because it is easy to understand. Datasets in the real world often represent more exciting events.