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# Probability Distribution

## Probabilities associated with possible counting number

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Visualizing Probability Distribution

Writing down all of the various probabilities of outcomes of an event is fine, but it can get a little tedious both to create and to read a long list of different probabilities. How else can we display the information from a probability distribution?

### Visualizing Probability Distributions

Probability distributions can convey a fantastic amount of useful information, but there may be so much information to view that the important points get lost in the data. Because of this, it is very common to create a graphical representation of the data to highlight important or interesting values.

Tables, histograms and bar charts in particular are excellent means of visualizing the data from discrete probability distributions. If you use a histogram or bar chart, by enumerating the various outcomes along the \begin{align*}x\end{align*}-axis and the expected probability of occurrence on the \begin{align*}y\end{align*}-axis, you create a very concise and easily read summary of the distribution of outcome probabilities.

#### Illustrating Probability Distributions

Let \begin{align*}C\end{align*} be a discrete random variable representing the number of heads that might result from flipping a coin three times. Create a bar chart to illustrate the probability distribution of \begin{align*}C\end{align*}.

Start by identifying the possible outcomes of flipping a coin three times:

\begin{align*}&\text{TTT has 0 heads.} \qquad \text{TTH has 1 heads.}\\ &\text{THT has 1 heads.} \qquad \text{THH has 2 heads.}\\ &\text{HTT has 1 heads.} \qquad \text{HTH has 2 heads.}\\ &\text{HHT has 2 heads.} \qquad \text{HHH has 3 heads.}\end{align*}

So we have 1 possibility with 0 heads: \begin{align*}P(0) = \frac{1}{8} = 0.125\end{align*}

\begin{align*}& 3\ \text{possibilities with} \ 1 \ \text{heads}: && P(1) = \frac{3}{8} = 0.375 \\ & 3 \ \text{possibilities with} \ 2 \ \text{heads}: && P(2) = \frac{3}{8} = 0.375 \\ & 1 \ \text{possibility with} \ 3 \ \text{heads}: && P(3) = \frac{1}{8} = 0.125\end{align*}

#### Creating Tables

Create a table showing the probability distribution of the possible outcomes of rolling two standard dice.

Let random variable \begin{align*}S\end{align*} represent the sum of the pips showing on the roll of both dice. We know then than \begin{align*}2 \le S \le 12\end{align*}.

Find all of the possible outcomes of rolling two dice, as shown in the image on the below.

Create a table showing the probabilities of each possible outcome of \begin{align*}S\end{align*}:

 \begin{align*}S\end{align*} \begin{align*}P(S)\end{align*} \begin{align*}S\end{align*} \begin{align*}P(S)\end{align*} 2 \begin{align*}\frac{1}{36}\end{align*} 8 \begin{align*}\frac{5}{36}\end{align*} 3 \begin{align*}\frac{2}{36}\end{align*} 9 \begin{align*}\frac{4}{36}\end{align*} 4 \begin{align*}\frac{3}{36}\end{align*} 10 \begin{align*}\frac{3}{36}\end{align*} 5 \begin{align*}\frac{4}{36}\end{align*} 11 \begin{align*}\frac{2}{36}\end{align*} 6 \begin{align*}\frac{5}{36}\end{align*} 12 \begin{align*}\frac{1}{36}\end{align*} 7 \begin{align*}\frac{6}{36}\end{align*}

#### Creating Probability Histograms

Create a probability histogram of the possible outcomes of rolling two dice. You may use your data from Example B.

In the previous example, we created a table of the probabilities of each outcome of rolling two dice, designated as discrete random variable \begin{align*}S\end{align*}. Let’s add one more column for each value so we can convert the fractional probability to decimal:

 \begin{align*}S\end{align*} \begin{align*}P(S)\end{align*} \begin{align*}P(S)\end{align*} decimal \begin{align*}S\end{align*} \begin{align*}P(S)\end{align*} \begin{align*}P(S)\end{align*} decimal 2 \begin{align*}\frac{1}{36}\end{align*} .028 8 \begin{align*}\frac{5}{36}\end{align*} .139 3 \begin{align*}\frac{2}{36}\end{align*} .056 9 \begin{align*}\frac{4}{36}\end{align*} .111 4 \begin{align*}\frac{3}{36}\end{align*} .083 10 \begin{align*}\frac{3}{36}\end{align*} .083 5 \begin{align*}\frac{4}{36}\end{align*} .111 11 \begin{align*}\frac{2}{36}\end{align*} .056 6 \begin{align*}\frac{5}{36}\end{align*} .139 12 \begin{align*}\frac{1}{36}\end{align*} .028 7 \begin{align*}\frac{6}{36}\end{align*} .167

We can use this data to create a histogram, setting the \begin{align*}y\end{align*}-axis to the probability and the \begin{align*}x\end{align*}-axis to the values of \begin{align*}S\end{align*}:

#### Earlier Problem Revisited

Writing down all of the various probabilities of outcomes of an event is fine, but it can get a little tedious both to create and to read a long list of different probabilities. How else can we display the information from a probability distribution?

Tables, histograms, bar graphs and pie charts are the most common visual representations of probability distributions.

### Examples

#### Example 1

Let  \begin{align*}R\end{align*} be a discrete random variable representing the number of red marbles pulled over three trials of pulling and replacing one marble out of a bag containing 4 red, 4 yellow, and 4 green marbles. Create a probability distribution table for \begin{align*}R\end{align*}.

The possible values of \begin{align*}R\end{align*} are 1, 2, and 3. There is a \begin{align*}\frac{1}{3}\end{align*} chance or red on each pull. The probability distribution for \begin{align*}R\end{align*} would thus be:

 \begin{align*}P(1)\end{align*} \begin{align*}.333\end{align*} \begin{align*}P(2)\end{align*} \begin{align*}(.333)^2=.111\end{align*} \begin{align*}P(3)\end{align*} \begin{align*}(.333)^2=.037\end{align*}

#### Example 2

Let \begin{align*}S\end{align*} be a discrete random variable representing the number of 2's you spin over 5 spins on a spinner with 4 equally-spaced points. Create a histogram showing the probability distribution of \begin{align*}S\end{align*}

The possible values of \begin{align*}S\end{align*} are 1, 2, 3, 4, and 5. There is a \begin{align*}\frac{1}{4}\end{align*} chance of a 2 on each spin:

#### Example 3

Create a probability distribution table for the outcomes of the sum of two 5-sided dice.

Let’s start by creating a grid to show all of the possible combinations:

 1 2 3 4 5 1 2 3 4 5 6 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10

Now we can create a distribution based on the probability of each possible outcome 2-10, let \begin{align*}R\end{align*} be a discrete random variable representing the sum of the dice:

 \begin{align*}R\end{align*} 2 3 4 5 6 7 8 9 10 \begin{align*}P(R)\end{align*} \begin{align*}\frac{1}{25} = .04\end{align*} \begin{align*}\frac{2}{25} =.08\end{align*} \begin{align*}\frac{3}{25} =.12\end{align*} \begin{align*}\frac{4}{25} =.16\end{align*} \begin{align*}\frac{5}{25} =.02\end{align*} \begin{align*}\frac{4}{25} =.16\end{align*} \begin{align*}\frac{3}{25} =.12\end{align*} \begin{align*}\frac{2}{25} =.08\end{align*} \begin{align*}\frac{1}{25} =.04\end{align*}

### Review

1. Create a probability distribution table for a single roll of two 7-sided dice.
2. Create a histogram to visualize the data from problem 1.
3. Create a pie chart showing the same data.
4. There are 12 green, 9 blue, and 4 red candies in an opaque bag. Let \begin{align*}R\end{align*} be a discrete random variable representing the number of red candies you get in a row by pulling and replacing one candy four times. Create a probability distribution table illustrating the possible outcomes of \begin{align*}R\end{align*}.
5. Create a histogram illustrating the information from problem 4.
6. Create a pie chart showing the same data.
7. Let discrete random variable \begin{align*}S\end{align*} represent the number of 7’s you get when rolling two 5-sided dice three times. Create a probability distribution table for \begin{align*}S\end{align*}.
8. Create a histogram illustrating the information from problem 7.
9. Create a pie chart with the same information.
10. Let \begin{align*}T\end{align*} be the number of tails you get when you flip a fair coin 4 times, create a probability distribution table for \begin{align*}T\end{align*}.
11. Create a histogram or bar chart for \begin{align*}T\end{align*}, from problem 10.
12. Create a pie chart for \begin{align*}T\end{align*} from problem 10.

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### Vocabulary Language: English

Histogram

A histogram is a display that indicates the frequency of specified ranges of continuous data values on a graph in the form of immediately adjacent bars.

probability distribution

In a probability distribution, you may have a table, a graph, or a chart that shows you all the possible values of X (your variable), and the probability associated with each of these values P(X).