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# Probability Distribution

## Probabilities associated with possible counting number

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Probability Distribution

#### Objective

In this lesson, you will learn about probability distributions and how they describe the probabilities associated with different possible values of a random variable.

#### Concept

Assume you have an unfair coin that is weighted to land on heads 65% of the time. If you flip that coin 3 times and let \begin{align*}T\end{align*} represent the number of tails you get, what is the probability distribution for \begin{align*}T\end{align*}?

Look to the end of the lesson for the answer.

#### Watch This

http://youtu.be/s2S1oD3ovps statslectures - Discrete Probability Distributions

#### Guidance

A probability distribution is a list of each value a random variable can attain, along with the probability of attaining each value. In other words, the probability distribution of an event is sort of a map of how each possible outcome relates to the chance it will happen.

For instance, the probability distribution of flipping a coin twice is:

If we define the random variable \begin{align*}X\end{align*} to be the number of heads you get when you flip a coin twice, we could create the following probability distribution table for \begin{align*}X\end{align*}

 \begin{align*}X\end{align*} \begin{align*}0\end{align*} \begin{align*}1\end{align*} \begin{align*}2\end{align*} \begin{align*}P(X)\end{align*} \begin{align*}\frac{1}{8}\end{align*} \begin{align*}\frac{3}{4}\end{align*} \begin{align*}\frac{1}{8}\end{align*}

There are various ways of visualizing a probability distribution, and we will review that concept in another lesson. For now, we focus on identifying what a probability distribution is, and how to calculate it for a particular event.

Example A

In Chi’s class, 4 students have one parent, 7 have two parents, and 1 student lives with his uncle. Let \begin{align*}P\end{align*} be the number of parents of a randomly selected student from the class. Create a probability distribution for \begin{align*}P\end{align*}.

Solution:

Set random variable \begin{align*}P\end{align*} to be the number of parents:

Now find the probability of each \begin{align*}P\end{align*}, noting that there are 12 students total:

Example B

Roll two fair six-sided dice. Let \begin{align*}D\end{align*} equal the sum of the dice. Create a probability distribution for \begin{align*}D\end{align*}.

Solution: Make a list of the individual probabilities of each of the 36 possible outcomes:

Example C

Janie wants to evaluate the probabilities of pulling various cards from a deck. She sets the discrete random variable \begin{align*}C\end{align*} to be the number of diamonds she gets over the course of three trials, if each trial consists of pulling, recording, and replacing one random card from a standard deck. What is the probability distribution of \begin{align*}C\end{align*}?

Solution: To evaluate the probability distribution of \begin{align*}C\end{align*}, Janie needs to identify the probability of each of the possible values of \begin{align*}C\end{align*}. Note that the chance she will pull a diamond is \begin{align*}\frac{13}{52}\end{align*} or \begin{align*}.25\end{align*}, meaning that the chance she will not pull a diamond is \begin{align*}1-.25=.75\end{align*}:

• For \begin{align*}C=(1)\end{align*}, the total probability is: \begin{align*}.14+.14+.14=.42 \ or \ 42 \%\end{align*} (see the three possible outcomes resulting in \begin{align*}C=1\end{align*} below)
• Diamond, other, other : \begin{align*}.25 \times .75 \times .75=.14\end{align*}
• Other, Diamond, other : \begin{align*}.75 \times .25 \times .75=.14\end{align*}
• Other, other, Diamond : \begin{align*}.75 \times .75 \times .25=.14\end{align*}
• For \begin{align*}C=(2)\end{align*}, the total probability is: \begin{align*}.047+.047+.047=.141 \ or \ 14.1 \%\end{align*}
• Diamond, Diamond, other : \begin{align*}.25 \times .25 \times .75=.047\end{align*}
• Diamond, other, Diamond : \begin{align*}.25 \times .75 \times .25=.047\end{align*}
• Other, Diamond, Diamond : \begin{align*}.75 \times .25 \times .25=.047\end{align*}
• For \begin{align*}C=(3)\end{align*}, the probability is : \begin{align*}.25 \times .25 \times .25=.016 \ or \ 1.6 \%\end{align*}
• Diamond, Diamond, Diamond: \begin{align*}.25 \times .25 \times .25=.016\end{align*}
##### Concept Problem Revisited

Assume you have an unfair coin that is weighted to land on heads 65% of the time. If you flip that coin 3 times and let \begin{align*}T\end{align*} represent the number of tails you get, what is the probability distribution for \begin{align*}T\end{align*}?

If each throw has a 65% chance of heads, then it has a 35% chance of tails:

• For \begin{align*}T=1\end{align*}, we could have THH, HTH, or HHT. Each of those has a \begin{align*}.35 \times .65 \times .65=.15\end{align*} chance of occurring, so \begin{align*}P(T=1)=.15 \times 3=.45 \ or \ 45 \%\end{align*}
• For \begin{align*}T=2\end{align*}, we could have TTH, THT, or HTH. Each has a \begin{align*}.35 \times .35 \times .65=.08\end{align*} chance, so \begin{align*}P(T=2)=.08 \times 3=.24 \ or \ 24 \%\end{align*}
• For \begin{align*}T=3\end{align*}, we could have only TTT, with a chance of \begin{align*}.35 \times .35 \times .35=.043 \ or \ 4.3 \%\end{align*}

#### Vocabulary

A probability distribution is a list of each value a random variable can attain, along with the probability of attaining each value.

#### Guided Practice

1. Create a probability distribution for number of heads when you flip a coin 3 times.
2. Let \begin{align*} C\end{align*} be the number of chocolate chip cookies you get if you randomly pull and replace two cookies from a jar containing 6 chocolate chip, 4 peanut butter, 8 snickerdoodle, and 12 sugar cookies. Create a probability distribution for \begin{align*}C\end{align*}.
3. Let \begin{align*}S\end{align*} be the score of a single student chosen at random from Mr. Spence’s class. Create a probability distribution for \begin{align*}S\end{align*}, given the following:
 Number of Students Test 11 87 7 89 13 92 9 94 6 96

Solutions:

1. Write out all the possibilities:

2. There are a total of 30 cookies, the probability of pulling a chocolate chip cookie is \begin{align*}\frac{6}{30}=.20\end{align*}, so the probability of not pulling a chocolate chip is \begin{align*}\frac{24}{30}=.80\end{align*}

• For \begin{align*}C=0\end{align*} we have to pull a non-chocolate chip both times: \begin{align*}.8 \times .8=.64 \ or \ 64 \%\end{align*}
• For \begin{align*}C=1\end{align*} we could either pull the chocolate chip cookie first or second, so we get \begin{align*}(.2 \times .8)+(.8 \times .2)=.32 \ or \ 32 \% \end{align*}
• For \begin{align*}C=2\end{align*} we have to pull chocolate chip both times, so we have \begin{align*}.2 \times .2=.04 \ or \ 4 \%\end{align*}

3. There are a total of 46 students in Mr. Spence’s class, so there are 46 scores. The probability of a random student having score \begin{align*}S\end{align*} is the same as that score’s portion of the total number of scores:

• \begin{align*}P(S=87)=\frac{11}{46}\end{align*}
• \begin{align*}P(S=89)=\frac{7}{46}\end{align*}
• \begin{align*}P(S=94)=\frac{9}{46}\end{align*}
• \begin{align*}P(S=96)=\frac{6}{46}\end{align*}

#### Practice

1. What is a probability distribution?

2. What is a random variable?

3. What is the difference between a discrete and a continuous random variable?

For problems 4-7, refer to the following table:

 \begin{align*}S\end{align*} 2 3 4 5 6 7 8 9 10 \begin{align*}P(S)\end{align*} 0.04 0.12 0.16 0.16 0.12 0.04

4. Assuming the table is a probability distribution for discrete random variable \begin{align*}S\end{align*}, which is the sum of two dice rolled once, how many sides does each die have?

5. What is \begin{align*}P(3)\end{align*}?

6. What is \begin{align*}P(6)\end{align*}?

7. What is \begin{align*}P(9)\end{align*}?

8. Roll two seven-sided dice once. Let \begin{align*}S\end{align*} be the sum of the two dice. Create a probability distribution for \begin{align*}S\end{align*}.

9. Flip a fair coin 3 times, let \begin{align*}H\end{align*} be the number of heads. Create a probability distribution for \begin{align*}H\end{align*}.

10. Let \begin{align*}S\end{align*} be the sum of two standard fair dice. Create a probability distribution for \begin{align*}S\end{align*}, if the experiment consists of a single roll of both dice.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.3.

### Vocabulary Language: English

Histogram

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

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).