PMF and CMF

Probability Director uses PMF and CMF to describe discrete probability results such as Dice totals, Success Counts, Binomial results, and other exact weighted outcomes.

You do not need a statistics background to use them.

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PMF — Probability Mass Function

A PMF answers:

What is the probability of this exact outcome?

For a standard 2D6 roll, the possible totals are:

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

The PMF tells you the probability of each total.

Example:

2D6 Total	Probability
2	2.78%
3	5.56%
4	8.33%
5	11.11%
6	13.89%
7	16.67%
8	13.89%
9	11.11%
10	8.33%
11	5.56%
12	2.78%

So:

PMF(7) = 16.67%

means:

The probability of rolling exactly 7 is about 16.67%.

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How to Think About a PMF

A PMF is basically a probability table.

Outcome -> Probability

For example:

2  -> 2.78%
3  -> 5.56%
4  -> 8.33%
...
12 -> 2.78%

If you want to know the probability of one exact discrete result, use the PMF.

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CMF — Cumulative Mass Function

A CMF answers:

What is the probability of getting this outcome or anything below it?

It adds the PMF probabilities together as you move through the outcomes.

For 2D6:

CMF(5)
=
P(2) + P(3) + P(4) + P(5)

Using the probabilities above:

CMF(5)
≈ 2.78% + 5.56% + 8.33% + 11.11%
≈ 27.78%

So:

CMF(5) = 27.78%

means:

There is about a 27.78% chance of rolling 5 or less.

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PMF vs. CMF

The easiest distinction is:

Question	Use
What is the chance of rolling exactly 7?	PMF
What is the chance of rolling 7 or less?	CMF
What is the chance of exactly 3 successes?	PMF
What is the chance of 3 or fewer successes?	CMF

In short:

PMF = exact outcome
CMF = cumulative probability up to that outcome

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Example

Suppose a game has this discrete result:

Outcome	PMF	CMF
1	10%	10%
2	20%	30%
3	30%	60%
4	25%	85%
5	15%	100%

PMF example

PMF(3) = 30%

Meaning:

The probability of getting exactly 3 is 30%.

CMF example

CMF(3) = 60%

Meaning:

The probability of getting 3 or less is 60%.

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Why the CMF Always Ends at 100%

The CMF keeps adding all possible outcomes.

By the time it reaches the maximum possible outcome, every possible result has been included.

So:

CMF(Maximum Outcome) = 100%

For 2D6:

CMF(12) = 100%

because every possible 2D6 total is 12 or less.

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The CMF Never Goes Down

A PMF can rise and fall:

2D6 PMF

2   low
3   higher
4   higher
...
7   highest
...
12  low

A CMF only stays the same or increases:

2   2.78%
3   8.33%
4  16.67%
5  27.78%
...
12 100%

That is because each new outcome adds more probability to the accumulated total.

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“At or Below” Uses the CMF

If your gameplay question is:

What is the chance of rolling 10 or less?

that is a cumulative question.

Conceptually:

P(X <= 10)

That is what the CMF gives you.

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“Above” Can Be Derived from the CMF

If you want:

What is the probability of rolling above 10?

you can use:

1 - CMF(10)

Example:

CMF(10) = 91.67%

P(X > 10)
=
100% - 91.67%
=
8.33%

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Range Probabilities

A CMF can also help calculate ranges.

Suppose you want:

What is the probability of rolling from 5 through 9?

Conceptually:

P(5 <= X <= 9)

Using cumulative probabilities:

CMF(9) - CMF(4)

This removes everything below 5 while keeping everything through 9.

Probability Director’s exact range APIs handle inclusive and exclusive endpoints directly, so you normally do not need to calculate this manually.

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PMF and Dice

Dice charts commonly use a PMF.

For 2D6, each bar answers:

How likely is this exact total?

Example:

Outcome 7
PMF = 16.67%

This is why a Dice PMF chart has one bar for each possible final total.

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PMF and Success Count

A Dice Pool Success Count also naturally uses a PMF.

Example:

10D6
Success on 5+

Possible results:

0, 1, 2, ... 10 successes

The PMF tells you:

P(exactly 4 successes)
P(exactly 5 successes)
P(exactly 6 successes)

The CMF tells you:

P(4 or fewer successes)
P(5 or fewer successes)
P(6 or fewer successes)

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PMF and Binomial

Binomial results work the same way.

Suppose:

10 trials
30% success probability

One Binomial sample produces:

0 ... 10 successes

The PMF answers:

What is the probability of exactly K successes?

The CMF answers:

What is the probability of K or fewer successes?

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PMF and Custom Weighted Outcomes

For a weighted numeric table:

Outcome	Weight
10	1
20	2
50	1

the PMF is:

Outcome	PMF
10	25%
20	50%
50	25%

The CMF is:

Outcome	CMF
10	25%
20	75%
50	100%

Notice that the outcomes do not need to be evenly spaced.

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What About Symbolic Outcomes?

A symbolic result such as:

Common
Rare
Epic
Legendary

has a PMF because each category can have an exact probability.

Example:

P(Rare) = 20%

But a cumulative function only makes mathematical sense when the outcomes have an explicit order.

Probability Director does not automatically assume:

Common < Rare < Epic < Legendary

just because those names look ordered to a human.

Without an authored ordering contract, symbolic outcomes should be treated as categories rather than cumulative numeric values.

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CMF vs. CDF

You may also see the term CDF:

Cumulative Distribution Function

For discrete results, the idea is the same:

P(X <= x)

In statistics, CDF is the more common general term.

A CMF can be thought of as the cumulative form of a discrete PMF.

For a developer, the practical distinction is simple:

PMF
= probability of exactly X

CMF / discrete CDF
= probability of X or less

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Quick Reference

PMF
Probability Mass Function

Question:
"What is the probability of exactly this outcome?"

Example:
P(X = 7)

CMF
Cumulative Mass Function

Question:
"What is the probability of this outcome or anything below it?"

Example:
P(X <= 7)

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One-Sentence Explanation

PMF tells you the chance of one exact discrete result; CMF tells you the total chance of that result and everything below it.