# Cumulative Distribution Function: Formula & Examples

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• 0:00 Dice Rolling Probability
• 0:45 Cumulative…
• 2:23 Solving for Range of Outcomes
• 3:33 Solving for Greater…
• 4:17 Lesson Summary

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Lesson Transcript
Instructor: James Walsh

M.B.A. Veteran Business and Economics teacher at a number of community colleges and in the for profit sector.

This lesson introduces you to the cumulative distribution function. An easy-to-follow illustration is used to show you the formula and it's usefulness. We also look at some examples.

## Dice Rolling Probability

Let's say your sister, Nikki, and her friend, Becky, are hanging out and playing board games. Like with many board games, they're playing a dice game that involves rolling the two dice and moving game pieces around a board. They're wondering as they play about the likelihood of rolling certain numbers when they throw the dice. By using probabilities, the percentages of rolling certain numbers can be precisely calculated. Nikki was looking around on the Internet and found a table that tells the chances that they will roll certain numbers. They look at the table, which you can see here:

Using the table, the probability that they would roll a five with two dice is 4/36 or 11.1%. They appreciate the table and decide to keep it out while they play.

## Cumulative Distribution Function

Becky has a problem as she gets ready to roll the dice. She needs to roll something less than six or she will land in a bad place on the board and have to pay Nikki rent. She wonders what her chances for success are. To determine that, she can use the cumulative distribution function.

The cumulative distribution function (FX) gives the probability that the random variable X is less than or equal to a certain number x. Its formula is:

for all R. R in a dice roll is the range of outcomes or {2...12}

In general, we can assume that the probability for anything less than two is zero, since you cannot roll a one with two dice. Also, that the probability for all of the numbers in a defined range will equal 1 or 100%.

So, back to Becky's problem, she needs to roll a five or less or she will be in trouble in the game. Using the cumulative distribution formula, her problem looks like this:

FX (5) = P(X <_ 5) that is the probability (P) that dice roll (X) is less than or equal to five (x).

She needs the dice to show a 2, 3, 4, or 5 to be safe. Using the table, she adds the ratios for 2, 3, 4, and 5, which are 1/36, 2/36, 3/36, and 4/36. She adds them together and gets 10/36 or 27.7%.

She thinks that's at least a little better than 1 in 4, so she'll just have to be brave and give them a roll!

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