# Probability density function of dice

• rhyno89
In summary, the conversation discusses finding the probability of getting a certain number of different faces when rolling a fair die three times. The probabilities for one, two, and three different faces are calculated and it is noted that they must sum to one. The correct combinations for each scenario are determined and the final probabilities are calculated.
rhyno89

## Homework Statement

Roll a fair die three times
Let X be the number of different faces shown all together ( X = 1,2,3 )
Find px(k)

## The Attempt at a Solution

Alright so I kno that i need to get the individual probabilities of each outcome
The first one where only one face is shown is (1/6)^3 * 6

The one where there are two different faces is (1/6)^2 * (5/6)

And I assume that likewise the final one with three different ones is (1*5*6)/(6^3)

Where I am getting stuck is I kno they have to sum to one so finding the different combinations to multiply to these expressions is giving me some trouble. For example the combinations in the same face was 6 becuase it could be either 1,2,3,4,5,6. Currently I have part 2 as having 30 different outcomes but it winds up looking a little off...any help would be great

I'll agree with you one the first one. The first die can be anything and then the other two have to match. So (1/6)*(1/6). I don't agree on the third one. Again the first die can be anything but now the second can be any of the five remaining sides and the third can be any of the four remaining. For the second case, you should first account for the fact that any of the three dice can be the different one, and then multiply by the probability that the other two are different from that, but the same as each other.

yea thanks i figured it out

The first one is 6(1/6^3)

The second is 6*5 * 3 different combos over 216

and the last one is 6*5*4/ 216

## 1. What is a probability density function (PDF)?

A probability density function (PDF) is a mathematical function that describes the probability of a random variable taking on a specific value or falling within a certain range of values. It is used to model the distribution of a continuous random variable, such as the outcomes of rolling dice.

## 2. How is the PDF of dice determined?

The PDF of dice is determined by the number of sides on the dice. For example, a standard six-sided die has a PDF that assigns equal probabilities (1/6 or 16.67%) to each of the possible outcomes (1, 2, 3, 4, 5, or 6). This is known as a uniform distribution.

## 3. Can the PDF of dice change?

Yes, the PDF of dice can change depending on the type of dice being used. For instance, a non-standard die with different numbers of sides or different probabilities assigned to each outcome will have a different PDF. Additionally, the PDF can change if the dice are biased or manipulated in some way.

## 4. How is the PDF of dice used in probability calculations?

The PDF of dice is used to calculate the probability of certain outcomes or ranges of outcomes. For example, if you roll two six-sided dice, the PDF can be used to determine the probability of getting a total sum of 7, which would be 6/36 or 1/6 (16.67%). The PDF can also be used to calculate the expected value and variance of a dice roll.

## 5. Why is understanding the PDF of dice important?

Understanding the PDF of dice is important for making accurate predictions and decisions based on probability. It allows us to calculate the likelihood of certain outcomes and make informed choices. Additionally, understanding the PDF can help us identify any biases or abnormalities in the dice, which can affect the fairness of games or experiments involving dice.

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